Minimal time sliding mode control for evolution equations in Hilbert spaces

TL;DR

This paper investigates minimal time sliding mode control for evolution equations in Hilbert spaces, providing optimality conditions and examples for complex systems like parabolic and reaction-diffusion equations.

Contribution

It introduces a novel approach to characterize minimal time sliding mode controllers in infinite-dimensional systems using the maximum principle.

Findings

Characterization of controllers via optimality conditions

Application to parabolic and reaction-diffusion systems

Framework covering broad class of evolution equations

Abstract

This work is concerned with the time optimal control problem for evolution equations in Hilbert spaces. The attention is focused on the maximum principle for the time optimal controllers having the dimension smaller that of the state system, in particular for minimal time sliding mode controllers, which is one of the novelties of this paper. We provide the characterization of the controllers by the optimality conditions determined for some general cases. The proofs rely on a set of hypotheses meant to cover a large class of applications. Examples of control problems governed by parabolic equations with potential and drift terms, porous media equation or reaction-diffusion systems with linear and nonlinear perturbations, describing real world processes, are presented at the end.

Equations

$$y^{\prime }(t)+Ay(t)=Bu(t),\text{mbox}t>0,$$

$$y(0)=y_0.$$

$$\text{mbox}Minimize\left\{J(T,u):=T;\text{mbox}(T,u)\in {\mathcal{U}}_{ad},\text{mbox}Py(T)=P{y}^{tar}\right\}$$

$${\mathcal{U}}_{ad}:=\{(T,u);\text{mbox}T\in \mathbb{R},\text{mbox}T>0,\text{mbox}u\in L^{\infty }(0,T;U),\text{mbox}{\Vert u(t)\Vert }_U\le \rho \text{mbox}a.e.t>0\}.$$

$$y_1^{\prime }(t)+(Ay(t){)}_1=B_1{u}_1(t),\text{mbox}{y}_2^{\prime }(t)+(Ay(t){)}_2=B_2{u}_2(t),\text{mbox}a.e.t>0$$

$$y_1^{\prime }(t)+(Ay(t){)}_1=B_1{u}_1(t),\text{mbox}{y}_2^{\prime }(t)+(Ay(t){)}_2=0,\text{mbox}a.e.t>0$$

$$V=V_1\times V_2,\text{mbox}H=H_1\times H_2,\text{mbox}{V}^{\ast }=V_1^{\ast }\times V_2^{\ast },\text{mbox}U=U_1\times U_2,\text{mbox}{U}^{\ast }=U_1^{\ast }\times U_2^{\ast }.$$

$$P:X=X_1\times X_2\to X,\text{mbox}P\in L(X,X),\text{mbox}$$

$$Py=(y_1,y_2)\text{mbox}orPy=(y_1,0),\text{mbox}fory=(y_1,y_2)\in X.$$

$$A^{\prime }(y)z=\underset{\lambda \to 0}{\lim }\frac{A(y+\lambda z)-Ay}{\lambda }\text{mbox}stronglyin{V}^{\ast },\text{mbox}forally,z\in V$$

$${⟨Ay-A\overline{y},y-\overline{y}⟩}_{V^{\ast },V}\ge {\alpha }_1{\Vert y-\overline{y}\Vert }_V^2-{\alpha }_2{\Vert y-\overline{y}\Vert }_H^2,\text{mbox}forally,\overline{y}\in V,\text{mbox}{\alpha }_1>0,$$

$$A\text{mbox}isboundedonboundedsubsetsofV,$$

$$D(A_H)=D({\Gamma }_H):=D_H.$$

$$(A_{H}y,{\Gamma }_{H}y{)}_H\ge {\alpha }_3{\Vert {\Gamma }_{H}y\Vert }_H^2-{\alpha }_4{\Vert y\Vert }_V^2,\text{mbox}forally\in D_H.$$

$$B\in L(U,H).$$

$${\Vert A^{\prime }(y)z\Vert }_{V^{\ast }}\le C{\Vert z\Vert }_V(1+C_1{\Vert y\Vert }_V^{\kappa }),\text{mbox}forally,z\in V,\text{mbox}some\kappa \in \mathbb{R},\text{mbox}\kappa \ge 0,$$

$${\Vert A_H^{\prime }(y)z\Vert }_H\le C{\Vert z\Vert }_{D_H}(1+C_1{\Vert y\Vert }_V^{\kappa }),\text{mbox}forally,z\in D_H,\text{mbox}\kappa \ge 0.$$

$${\Vert A_H^{\prime }(y_n)\psi -A_H^{\prime }(y)\psi \Vert }_H$$

$$\text{mbox}as{y}_n$$

$${⟨(A^{\prime }(y){)}^{\ast }z,{\Gamma }_{\nu }z⟩}_{V^{\ast },V}\ge {\gamma }_1{\Vert {\Gamma }_{\nu }z\Vert }_H^2-{\gamma }_2{\Vert z\Vert }_V^2(1+{\gamma }_3{\Vert y\Vert }_V^l)-{\gamma }_4,\text{mbox}$$

$${\Vert P{\Gamma }_H^{-\alpha /2}v\Vert }_H\le C{\Vert B^{\ast }v\Vert }_{U^{\ast }},\text{mbox}v\in H\text{mbox}.$$

$$\left((A_H^{\prime }(y\right){)}^{\ast }v,{\Gamma }_H^{-\alpha }v{)}_H\ge C_1{\Vert {\Gamma }_H^{(1-\alpha )/2}v\Vert }_H^2-C_2{\Vert v\Vert }_H^2(1+C_3{\Vert y\Vert }_V^l),$$

$$\left((A_H^{\prime }(y\right){)}^{\ast }v,{\Gamma }_H^{-1}v{)}_H\ge C_1{\Vert v\Vert }_H^2-C_2{\Vert v\Vert }_{V^{\ast }}^2(1+C_3{\Vert y\Vert }_V^l),\text{mbox}$$

$${\Vert Pv\Vert }_{V^{\ast }}\le C^{\ast }{\Vert B^{\ast }v\Vert }_{U^{\ast }},\text{mbox}forv\in H.$$

$$\hat{y}(t)=(y_1^{\text{mbox}target},\hat{z})$$

$${⟨Ay(t)-A\hat{y},P(y(t)-\hat{y})⟩}_{V^{\ast },V}\ge -C_3{\Vert P(y(t)-\hat{y})\Vert }_H^2,\text{mbox}$$

$$\rho >{\rho }_1,\text{mbox}{\rho }_1:=C^{\ast }{\Vert A_H\hat{y}\Vert }_V.$$

$$\rho >{\rho }_1,\text{mbox}{\rho }_1:=C^{\ast }{\Vert A_H{y}^{tar}\Vert }_V,$$

$${⟨A^{\prime }(y)z,z⟩}_{V^{\ast },V}\ge {\alpha }_1{\Vert z\Vert }_V^2-{\alpha }_2{\Vert y\Vert }_H^2,\text{mbox}forallz\in V.$$

$$(A_H^{\prime }(y)z,z{)}_H\ge {\alpha }_1{\Vert z\Vert }_V^2-{\alpha }_2{\Vert y\Vert }_H^2,\text{mbox}forally,\text{mbox}z\in D_H.$$

$${\Vert A^{\prime }(y)\Vert }_{L(V,V^{\ast })}\le C(1+C_1{\Vert y\Vert }_V^{\kappa }).$$

Full text

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Minimal time sliding mode control for evolution equations in Hilbert spaces

Gabriela Marinoschi

“Gheorghe Mihoc-Caius Iacob”Institute of Mathematical Statistics and

Applied Mathematics of the Romanian Academy,

Calea 13 Septembrie 13, Bucharest, Romania

gabriela.marinoschi@acad.ro

Abstract. This work is concerned with the time optimal control problem for evolution equations in Hilbert spaces. The attention is focused on the maximum principle for the time optimal controllers having the dimension smaller that of the state system, in particular for minimal time sliding mode controllers, which is one of the novelties of this paper. We provide the characterization of the controllers by the optimality conditions determined for some general cases. The proofs rely on a set of hypotheses meant to cover a large class of applications. Examples of control problems governed by parabolic equations with potential and drift terms, porous media equation or reaction-diffusion systems with linear and nonlinear perturbations, describing real world processes, are presented at the end.

Mathematics Subject Classification. 2010. 35B50, 47H06, 47J35, 49K20, 49K27

Key words. Time optimal control, optimality conditions, sliding mode control, evolution equations, maximum principle, reaction-diffusion systems

1 Problem presentation

The purpose of this paper is to study the time optimal control for a family of evolution equations in Hilbert spaces. In time optimal control the optimality criterion is the elapsed time. Here, by the time optimal control problem we mean to search for a constrained internal controller able to drive the trajectory of the solution from an initial state to a given target set in the shortest time, while controlling over the complete timespan.

Minimum time control problems have been initiated by Fattorini in the paper [12] and developed later in the monograph [13]. A list of only few titles dealing with this subject, in special for problems governed by parabolic type equations includes [16], [17], [18], [20], [21], [23]. In what concerns problems governed by abstract evolution equations, we cite [2] and the monographs [3], [4], [7]. In [2] the existence and uniqueness of a viscosity solution was provided for the Bellman equation associated with the time-optimal control problem for a semilinear evolution equation in a Hilbert space, while in [5] the time optimal control was studied for the Navier-Stokes equations. The existence of the optimal time control for a phase-field system was proved in [20] for a regular double-well potential, by using the Carleman inequality and the maximum principle was established by using two controls acting in subsets of the space domain. The asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge, in an appropriate sense, to a given strictly negative operator was studied in [19]. For a large class of problems and aspects related to this subject we refer the reader to the recent monograph [22].

From the perspective of applications, many processes in engineering, physics, biology, medicine, environmental sciences, ecology require solutions relying on time optimal control problems. The theoretical results in this paper aim to cover models governed by parabolic equations with potential and drift terms and various reaction-diffusion systems with linear or nonlinear perturbations, or nonlocal control problems, presented in the last section.

Especially of interest in applications is to control a system using a controller whose dimension is smaller than that of the state system. In this case the initial datum is steered not into a point, but into a linear manifold of the state space, situation which is relevant for the sliding mode control (for some references see e.g., [8], [9], [11]). The solution to such a problem, which is more challenging from the mathematical point of view, is a central point in our theoretical approach.

We prove the existence of the time optimal control and the first order necessary conditions of optimality in relation with the evolution equation on a Hilbert space $H,$

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Here, $A$ is a nonlinear and unbounded operator over a Hilbert space $H,$ $B$ is a linear operator from a Banach space $U$ to $H,$ $u$ is a controller constrained to belong to $U$ and $y$ is the solution to (1.1) corresponding to the initial datum $y_{0}$ and controlled by $Bu.$ The following minimization problem is studied:

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where $y$ is the solution to (1.1)-(1.2) and

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As it will be further explained, $P$ is an algebraic projection of the solution from $H$ to $H$ or to a subspace of it, $y^{tar}$ is a prescribed target for the state and $\rho$ is a positive constant at our choice.

Relying on certain hypotheses ensuring the well-posedness of (1.1)-(1.2) and on the hypothesis of a not empty admissible set $\mathcal{U}_{ad},$ the existence of optimal controls is proved. A maximum principle is first provided for an intermediate approximating problem. This generates a sequence of approximating optimal solutions which converges to a precise optimal pair to $(\mathcal{P}),$ whose characterization is a central point of the paper.

The results are more relevant in the case of states with many components. That is why, for the sake of a clearer explanation and for a simpler notation, let us first assume that the state $y$ in (1.1)-(1.2) has two components, $y=(y_{1},y_{2})\in H=H_{1}\times H_{2}$ and $u=(u_{1},u_{2})\in U=U_{1}\times U_{2},$ where $H_{i}$ are Hilbert spaces and $U_{i}$ are Banach spaces, $i=1,2.$

We focus on two problems. The more challenging case is to steer, by the action of only one control $Bu=(B_{1}u_{1},0)$, only the first component $y_{1}(t)$ of the state $y(t)$ from its initial value into a manifold $\mathcal{S},$ within a minimal time $T^{\ast}$. In this case, the target manifold is $y_{1}=y_{1}^{\mbox{target}}.$ This action may be realized using effectively one controller acting in the first equation. Thus, the state is forced to reach the manifold $\mathcal{S}=\{y;$ $y_{1}=y_{1}^{\mbox{target}}\}$ on which it may continue to slide, for $t\geq T^{\ast},$ possibly under supplementary conditions and by performing a controller slight modification after the time $T^{\ast}$. This turns out to be in fact the sliding mode control and it will be detailed for a reaction-diffusion model in Section 6, Example 3.

Another possibility is to control both state components, forcing them to reach a prescribed point target $y^{tar}:=(y_{1}^{\mbox{target}},y_{2}^{\mbox{target}}),$ by employing two controllers, with $Bu=(B_{1}u_{1},B_{2}u_{2}).$

Because the intention is to simultaneously prove the objectives stated before, these are formalized by means of the minimization problem $(\mathcal{P})$ involving a mapping $P\in L(H,H)$ covering each of the following situations:

$(i)$ $P(y_{1},y_{2})=(y_{1},y_{2})\in H_{1}\times H_{2},$ $B(u_{1},u_{2})=(B_{1}u_{1},B_{2}u_{2}),$ $y^{tar}=(y_{1}^{\mbox{target}},y_{2}^{\mbox{target}})=Py^{tar},$ in the case when both state components are controlled by two controllers;

$(ii)$ $P(y_{1},y_{2})=(y_{1},0),$ $y_{1}\in H_{1},$ $y_{2}\in H_{2},$ $B(u_{1},u_{2})=(B_{1}u_{1},0),$ $Py^{tar}:=(y_{1}^{\mbox{target}},0),$ in the situation when only the first component is controlled by one controller.

With this notation, (1.1) can be rewritten

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in the case $(i)$ and

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in the case $(ii),$ where $Ay(t)=((Ay(t))_{1},(Ay(t))_{2}).$

Actually, in case $(ii),$ $P$ is an algebraic projection of $H$ into $H_{1}\times\{0\}\subset H$, mapping $y=(y_{1},y_{2})$ into its first component $y_{1}$ which is the only one controlled in this case. For the second component of $Py$ corresponding to the second equation which is not controlled we set the value zero, in order to ensure a compact notation consistent in the calculations (e.g., of the type $\left\|Pv\right\|_{H}\leq C\left\|Bv\right\|_{H})$ with the value $Bu_{2}=0.$

In both cases we agree to use the same notation $y^{tar}$ in order to allow a compact writing. In the first case, $y^{tar}$ contains the targets for each state component. In the second case, the essential role is played by the first component of $Py^{tar}$ while the second component of $y^{tar}$ plays no role. We can set the latter zero, even if this is not a target, because its action, as well as that of the second component of $Py$ will be cancelled in the calculations by the second zero component of $Bu$.

This explanation can be extended to the case with the state $y$ having $n$ components, either when all $n$ components are controlled by $n$ controllers, or when only $k$ trajectories $(y_{1},...,y_{k})$ are led into $(y_{1}^{\mbox{target}},...,y_{k}^{\mbox{target}}),$ by using $k$ controllers, via $Bu=(B_{1}u_{1},...B_{k}u_{k},0,...,0).$ We note that we can change the notation, indicating the vector $(y_{1},...,y_{k})$ of the first $k$ components still by $y_{1}$ and the vector $(y_{k+1},...,y_{n})$ by $y_{2}$ and we can use a similar notation for $u$ and $B.$ So, the general case can be reduced to that with two state components. To conclude, for the writing simplicity, we shall refer in the sequel to the case with two state components.

The paper is organized as follows. The theoretical results rely on a set of hypotheses, $(a_{1})-(a_{6}),$ $(b_{1}),$ $(c_{1}),$ listed in Section 2. For the passing to the limit proof in Theorem 5.5, Section 5.3, there are necessary some technical assumptions $(d_{1})-(d_{5}),$ including the hypothesis (2.14). This is essential for the characterization of the controller if only one state is controlled ($P\neq I,$ $Bu=(B_{1}u_{1},0)),$ which may be the more relevant in applications. This is one of the novelty of this paper, besides the results characterizing the controller for evolution equations with some general nonlinear operators. Section 3 includes some results of existence, beginning with the well-posedness of the state system (1.1)-(1.2), in Theorem 3.2. The existence of the minimum time is provided in Theorem 3.3. In Section 4, we employ an approximating problem $(\mathcal{P}_{\varepsilon})$ indexed along a small parameter $\varepsilon$ occurring in some penalization terms of the functional $J.$ After giving a basic result in Theorem 4.1 for the existence of a solution to $(\mathcal{P}_{\varepsilon}),$ the convergence of $(\mathcal{P}_{\varepsilon})$ to $(\mathcal{P})$ is proved in Theorem 4.2. This result is strong by asserting that if one fix an optimal pair $(T^{\ast},u^{\ast})$ in $(\mathcal{P}),$ the sequence of optimal pairs in $(\mathcal{P}_{\varepsilon})$ tends exactly to $(T^{\ast},u^{\ast})$. The necessary conditions of optimality for $(\mathcal{P}_{\varepsilon})$ are determined in Proposition 5.4 at the end of an extremely technical procedure, while in Theorem 5.5 the necessary conditions of optimality for $(\mathcal{P})$ are obtained as the limit of the previous ones, as $\varepsilon\rightarrow 0,$ after sharp estimates for the approximating solution. A particular case for $U$, usually encountered, is treated in Corollary 5.6. Applications of these results, including a detailed example of minimum time sliding mode control, are presented in the last section. In the Appendix we provide some definitions and general results necessary in the paper.

2 Functional framework and basic hypotheses

Functional framework.

Let $V_{i}$ and $H_{i},$ $i=1,2,$ be Hilbert spaces and consider the standard triplet $V_{i}\subset H_{i}\equiv H_{i}^{\ast}\subset V_{i}^{\ast},$ with compact embeddings, where $V_{i}^{\ast}$ is the dual of $V_{i}.$ Let $U_{i},$ $i=1,2,$ be Banach spaces with the duals $U_{i}^{\ast}$ uniformly convex, implying that $U_{i}^{\ast}$ and $U_{i}$ are reflexive (see e.g., [6], p. 2). Let us denote

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We recall that the operator $P$ was defined in the introduction (see $(i)-(ii)),$ but as a matter of fact we can define it on any space $X=X_{1}\times X_{2},$ where $X_{i}$ can be $V_{i},$ $H_{i},$ $V_{i}^{\ast}.$ Also, we use the same symbol $P$ for $X_{i}=U_{i}.$ Thus,

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and it is defined as

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It can be easily seen $P^{2}=P$ and $\left\|Py\right\|_{X}\leq\left\|y\right\|_{X}.$

Let $A:V\rightarrow V^{\ast}.$ We denote by $A_{H}:D(A_{H})\subset H\rightarrow H$ the restriction of $A$ on $H$ defined by $A_{H}y=Ay$ for $y\in D(A_{H})=\{y\in V;$ $Ay\in H\}.$ In the sequel, $\Gamma:V\rightarrow V^{\ast}$ is the duality mapping from $V$ to $V^{\ast}$ and $\Gamma_{H}$ is the restriction of $\Gamma$ to $H$ (see 7.15).

Let $A\in C^{1}(V,V^{\ast}).$ The Gâteaux derivative of $A$ is the linear operator $A^{\prime}(y):V\rightarrow V^{\ast}$ defined by

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and $A_{H}^{\prime}(y)$ is the restriction of $A^{\prime}(y)$ to $H$ (see (7.1)-(7.2)). Properties of these operators are given in the Appendix.

Notation. Let $X$ and $Y$ be Banach spaces. By $L^{p}(0,T;X)$ we denote the space of $p$-summable functions from $(0,T)$ to $X$, for $1\leq p\leq\infty.$ $W^{1,p}(0,T;X)=\{f;$ $f:[0,T]\rightarrow X,$ absolutely continuous, $f,$ $df/dt\in L^{p}(0,T;X)\}.$ $C(X,Y)$ and $C^{1}(X,Y)$ are the spaces of continuous and differentiable Gâteaux, respectively, operators from $X$ to $Y$. $L(X,Y)$ is the space of linear continuous operators from $X$ to $Y.$ We denote the scalar product and norm in the space $X$ by $(\cdot,\cdot)_{X}$ and $\left\|\cdot\right\|_{X},$ respectively.

We shall denote by $C,$ $C_{i},$ $\alpha_{i},$ $\gamma_{i},$ $i=0,1,2,...$ positive constants that may change from line to line.

Some other notation and definitions related to the hypotheses below can be found in the Appendix.

Hypotheses $(a_{1}),(a_{2}),(b_{1}),(c_{1})$

$(a_{1})$ $A:V\rightarrow V^{\ast}$ is demicontinuous, $A0=0,$

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$(a_{2})$

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$(b_{1})$

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$(c_{1})$ For each $y^{tar}\in H,$ and $\rho$ positive large enough, there exists** $T_{\ast}>0$** and $u\in L^{\infty}(0,T;U)$ with $\sup\nolimits_{t\geq 0}\left\|u(t)\right\|_{U}\leq\rho,$ such that $Py(T_{\ast})=Py^{tar},$ where $y$ is the solution to (1.1)-(1.2).

Hypotheses $(a_{1}),$ $(a_{2}),$ $(b_{1})$ are necessary to prove the state system well-posedness and the existence of the solution to $(\mathcal{P})$, in Section 3. The minimization problem $(\mathcal{P})$ is relevant if the set $\mathcal{U}_{ad}$ is not empty. Hypothesis $(c_{1})$ ensures that $\mathcal{U}_{ad}\neq\varnothing$ and it is used in the proof of the control existence. We specify that the proof of the controllability of (1.1)-(1.2) is beyond the objective of this paper. However, for the reader convenience, the existence of a least a pair $(T,u)$ in the admissible set, or equivalently an example of proving the controllability of (1.1)-(1.2) in some cases, is given in Appendix, Proposition 7.1. Next, in the Examples, the reliability of $(c_{1})$ is commented in each case.

Hypotheses $(a_{3})-(a_{6})$

$(a_{3})$ $A\in C^{1}(V;V^{\ast})$ and $A_{H}\in C^{1}(D_{H};H).$

$(a_{4})$ $A^{\prime}(y)$ and $A_{H}^{\prime}(y)$ defined by (7.1) and (7.2) respectively, satisfy

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$(a_{5})$ $A^{\prime}$ is strongly continuous from $V$ to $L_{s}(V,V^{\ast}),$ (see (7.3)) and $A_{H}^{\prime}(y)$ is strongly continuous from $V$ to $L(D_{H},H),$ namely

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$(a_{6})$ The adjoint operator $(A^{\prime}(y))^{\ast}:V\rightarrow V^{\ast}$ satisfies the condition

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for all $y,z\in V,$ some $l\geq 0,$ $\gamma_{1},\gamma_{2}>0,$ where $\Gamma_{\nu},$ $\nu>0,$ is the Yosida approximation of $\Gamma,$ see (7.17).

Hypotheses $(a_{3})-(a_{6})$ are necessary for the proof of the existence of the system in variations, the adjoint system and the determination of the approximating optimality conditions.

Hypotheses $(d_{1})-(d_{5})$

Assume that $U\neq H$ and that there exists $\alpha\in(0,1)$ such that

$(d_{1})$

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$(d_{2})$ $(A_{H}^{\prime}(y))^{\ast}$ satisfies the relations

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for all $y,v\in D_{H},$ some $l\geq 0,$ and

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$(d_{3})$

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$(d_{4})$ Let $Py=(y_{1},0)$ and let $\rho$ be sufficiently large. For each $u\in L^{\infty}(0,\infty;U),$ with $\left\|u(t)\right\|_{U}\leq\rho$ there exists $\widehat{z},$ possibly depending on $u,$ such that

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satisfies

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for all $y\in V,$ and $t\in(0,T_{\ast}+\delta),$ with the choice

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$(d_{5})$ Let $P=I,$ assume

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and relation (2.14), where $\widehat{y}$ is replaced by $y^{tar}=(y_{1}^{\mbox{target}},y_{2}^{\mbox{target}}).$

We specify that $C^{\ast}$ in (2.15) and (2.16) is exactly the constant $C^{\ast}$ occurring in (2.12), depending on the domain $\Omega$ and $B,$ $T_{\ast}$ is the time specified in the controllability hypothesis $(c_{1}),$ $\delta$ is arbitrary and $y(t)$ is the solution to (1.1)-(1.2) corresponding to $u.$

Assumption (2.14) is a basic statement in the proof of the characterization of the controller in the case when only one state is controlled by one controller. This is the case when the state is allowed to reach a sliding manifold.

We also note that if $B_{1}=B_{2}=I$ or $B_{1}=I$ and $B_{2}=0$ and the spaces are such that $V\subset U$ or $H\subset U,$ then (2.12) is automatically satisfied. The case $U=H$ will be treated in Corollary 5.6.

Immediate consequences of the previous hypotheses are:

The operator $\lambda I+A,$ for $\lambda$ positive large enough, is coercive and $A_{H}$ is quasi $m$-accretive on $H\times H,$ implied by $(a_{1}).$

The operator $A^{\prime}(y)$ satisfies the estimate

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The operator $\left.A^{\prime}(y)\right|_{H}=A_{H}^{\prime}(y)$ is quasi $m$-accretive for each $y\in V$ and

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By (2.5) we have

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3 Existence results

In this section we provide the proofs of the existence of the solution to the state system and of a solution to the minimization problem $(\mathcal{P}).$ All over in this section, we assume $(a_{1}),$ $(a_{2}),$ $(b_{1}),$ $(c_{1}).$ Let

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**Definition 3.1. **A strong solution to the Cauchy problem (1.1)-(1.2) is a continuous function $y:[0,T]\rightarrow H,$ which is a.e. differentiable and satisfies (1.1) a.e. $t\in(0,T)$ and (1.2).

Theorem 3.2. Let $T>0,$ $u\in L^{2}(0,T;U),$ $y_{0}\in V.$* Then, *(1.1)-(1.2) has a unique strong solution $y\in X_{T},$ satisfying

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with $C$ a positive constant. Moreover, for two solutions $y$ and $\overline{y}$ corresponding to $u$ and $\overline{u}$ we have

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Finally, if $u_{n}\in L^{\infty}(0,T;U),$ $u_{n}\rightarrow u$ weak-star in $L^{\infty}(0,T;U),$ then the solution $y_{n}$ corresponding to $u_{n}$ tends to $y,$ the solution corresponding to $u,$ namely

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Proof. We recall that $A_{H}$ is quasi $m$-accretive on $H\times H.$ Assume first that the right-hand side of (1.1), $f=Bu$ is in $W^{1,1}(0,T;H)$ and $y_{0}\in D_{H}=D(A_{H})$. In this case we obtain a unique solution $y\in W^{1,\infty}(0,T;H)\cap L^{\infty}(0,T;D_{H})$ (see e.g., [6], p. 151, Theorem 4.9), implying that $A_{H}y\in L^{\infty}(0,T;H).$ A first estimate is obtained by testing the equation (1.1) by $y(t)$ and integrating it over $(0,t),$

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which yields

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Then, we multiply (1.1) in $H$ by $\Gamma_{H}y(t),$ use (2.4) and integrate over $(0,t),$ obtaining

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Using (3.5) we get

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for all $t\in[0,T],$ $T>0.$ We note that $C_{T}$ is continuous and increasing with respect to $T,$ but it can vary from line to line via the constant $C.$ This implies that

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By comparison with (1.1) we deduce that $\int_{0}^{T}\left\|y^{\prime}(\tau)\right\|_{H}^{2}d\tau\leq C_{T}.$ By gathering all estimates we obtain (3).

To prove (3.3) we consider two solutions corresponding to $(y_{0},u)$ and $(\overline{y_{0}},\overline{u}),$ write the difference of the equations for these solutions, test it by $(y-\overline{y})(t),$ integrate over $(0,t)$ and apply the Gronwall lemma.

We proceed further by a density argument. We take $y_{0}^{n}\in D_{H}$ and $u_{n}\in W^{1,1}(0,T;U)$ such that $y_{0}^{n}\rightarrow y_{0}$ strongly in $V$ and $u_{n}\rightarrow u$ strongly in $L^{2}(0,T;U),$ the latter implying $Bu_{n}\rightarrow Bu$ strongly in $L^{2}(0,T;H).$ It follows that the solution to (1.1) with $Bu_{n}$ instead of $Bu$ and with the initial datum $y_{0}^{n}$ has a unique strong solution $y_{n}\in W^{1,\infty}(0,T;H)\cap L^{\infty}(0,T;D_{H}),$ satisfying (3) and (3.3). From here, it follows that $y_{n}\rightarrow y$ strongly in $C([0,T];H)\cap L^{2}(0,T;V),$ as $n\rightarrow\infty,$ and the estimate (3.3) for $y_{n}$ is preserved at limit. The right-hand side of (3) is bounded and so $A_{H}y_{n}\rightarrow A_{H}y$ weakly in $L^{2}(0,T;H),$ since $A_{H}$ is strongly-weakly closed, and $y_{n}^{\prime}\rightarrow y^{\prime}$ weakly in $L^{2}(0,T;H).$ The estimate (3) is preserved at limit by the lower weakly continuity of the norms.

Let $u_{n}\in L^{\infty}(0,T;U),$* $u_{n}\rightarrow u$ weak-star in $L^{\infty}(0,T;U).$ *Then, (1.1)-(1.2) has a unique solution $y_{n}$ satisfying (3). Since the estimates are uniform, on a subsequence we get the convergences in the first line of (3.4). The strongly convergence follows by the Aubin-Lions lemma and the last one by the Arzelà-Ascoli theorem. Passing to the limit in (1.1)-(1.2) written for $y_{n}$ we get (1.1)-(1.2) corresponding to $y.$ $\square$

We observe that by (3.4) we deduce that** $y\in C_{w}([0,T];V),$** that is, except for a subset of zero measure, $y$ is a weak continuous function from $[0,T]$ in $V.$ We recall that $y\in C_{w}([0,T];V),$ if when $t_{n}\rightarrow t,$ as $n\rightarrow\infty,$ it follows that $y(t_{n})\rightarrow y(t)$ weakly in $V.$ Indeed, in the proof of Theorem 3.2 we have for $t_{n}\rightarrow t,$ that $y(t_{n})\rightarrow y(t)$ strongly in $H.$ On the other hand, $\left\|y(t_{n})\right\|_{V}\leq C_{T}$ and so $y(t_{n})\rightarrow\xi$ weakly in $V.$ But the limit is unique and so $\xi=y(t)\in V,$ for all $t\geq 0.$

Similarly, we deduce that $Ay\in C_{w}([0,T];V^{\ast}).$ By (2.2), $\left\|Ay(t_{n})\right\|_{V^{\ast}}$ is bounded, since $\left\|y(t_{n})\right\|_{V}\leq C_{T}$ and so $Ay(t_{n})\rightarrow\zeta$ weakly in $V^{\ast}.$ On the other hand, $Ay(t_{n})\rightarrow Ay(t)$ strongly in  $D_{H}^{\ast},$ the dual of $D_{H}$ because $y(t_{n})\rightarrow y(t)$ in $H.$ Thus, $\zeta=Ay(t)\in V^{\ast},$ for all $t\geq 0.$

Now we prove the existence of the minimum in $(\mathcal{P}).$ Recall that $y^{tar}:=(y_{1}^{\mbox{target}},y_{2}^{\mbox{target}})$ if $P=I$ and $y^{tar}:=(y_{1}^{\mbox{target}},z),$ $z\in H$ if $P\neq I.$

Theorem 3.3. Let

[TABLE]

Then, problem $(\mathcal{P})$ has at least one solution $(T^{\ast},u^{\ast})$ with the corresponding state $y^{T^{\ast},u^{\ast}}:=y^{\ast}.$

Proof. We recall that we have assumed $(c_{1})$ asserting that the admissible set $\mathcal{U}_{ad}\neq\varnothing.$ The functional $J(T,u)=T$ is nonnegative, hence it has an infimum. We denote $\inf J(T,u)=T^{\ast}\geq 0.$ Let us consider a minimizing sequence $(T_{n},u_{n}),$

[TABLE]

where $y^{T_{n},u_{n}}$ is the solution to the state system corresponding to $(T_{n},u_{n}),$ such that

[TABLE]

On a subsequence it follows that

[TABLE]

We see that $T_{n}\rightarrow T^{\ast}$ and passing to the limit in (3.8) we get that $J(T^{\ast},u^{\ast})=T^{\ast}.$ The state system corresponding to each $T>0$ and $u_{n}$ has a unique solution $y^{T,u_{n}}$ satisfying (3). In particular, this is true for $T=T_{n}$ and $T=T^{\ast}.$ We note that the restriction of the solution $y^{T_{n},u_{n}}$ to $(0,T^{\ast})$ is in fact the solution $y^{T^{\ast},u_{n}}.$ We have by (3)

[TABLE]

where $C_{T}$ depends continuously and increasingly on $T$ (see (3)). Therefore, by selecting a subsequence and recalling (3.4) we have

[TABLE]

[TABLE]

since $A_{H}$ is strongly-weakly closed. By Ascoli-Arzelà theorem we still get

[TABLE]

Also, by the last assertion in Theorem 3.2 we infer that $y^{\ast}$ is the solution to the state system corresponding to $(T^{\ast},u^{\ast}).$ We show next the convergence of $y^{T_{n},u_{n}}$ to $y^{\ast}$ as $n\rightarrow\infty.$ For any $v\in H$ we have

[TABLE]

because $\left\|y^{T_{n},u_{n}}(t)\right\|_{L^{\infty}(0,T_{n};H)}\leq C_{T_{n}}\leq C_{T^{\ast}+1}$ and by (3.4). We took into account that $y^{T_{n},u_{n}}$ restricted to $(0,T^{\ast})$ coincides with $y^{T^{\ast},u_{n}.}$ In a similar way, we can prove the weak convergences of the other sequences, that is $(y^{T_{n},u_{n}})^{\prime}\rightarrow(y^{\ast})^{\prime},$ $A_{H}y^{T_{n},u_{n}}\rightarrow A_{H}y^{\ast}$ weakly in $L^{2}(0,T^{\ast};H)$ and that $y^{T_{n},u_{n}}\rightarrow y^{\ast}$ strongly in $L^{2}(0,T^{\ast};H).$

It remains to prove that $Py^{\ast}(T^{\ast})=Py^{tar}.$ We have

[TABLE]

which tend to zero since $\left\|P(y^{T_{n},u_{n}})^{\prime}\right\|_{L^{2}(0,T_{n};H)}\leq C_{T^{\ast}+1}$ and by (3.4). Hence

[TABLE]

From here, we also deduce that $T^{\ast}>0.$ Otherwise, we would have $Py^{\ast}(T^{\ast})=Py^{\ast}(0)=Py_{0},$ that is $Py_{0}=Py^{tar}$ which contradicts the hypothesis that $Py_{0}\neq Py^{tar}.$ Thus, we have obtained $T^{\ast}>0,$ $u^{\ast}$ with the restriction $\left\|u^{\ast}\right\|_{L^{\infty}(0,T^{\ast};U)}\leq\rho$, and $J(T^{\ast},u^{\ast})=T^{\ast}.$ We have got that $T^{\ast}$ is the unique infimum time at which $Py^{\ast}(T^{\ast})=Py^{tar}.$ This ends the proof. $\square$

4 The approximating problem

Let $\varepsilon$ be positive and consider the problem

[TABLE]

subject to (1.1)-(1.2), where

[TABLE]

**Theorem 4.1. **Let $y_{0}\in V,$ $Py^{tar}\in P(H),$ $Py_{0}\neq Py^{tar}.$ Then, problem $(\mathcal{P}_{\varepsilon})$ has at least a solution $(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast})$, with the corresponding state $y^{T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast}}:=y_{\varepsilon}^{\ast}.$

**Proof. **Since $J_{\varepsilon}(T,u)$ is nonnegative, there exists $d_{\varepsilon}=\inf J_{\varepsilon}(T,u)$ and it is positive. Indeed, we note that if $J_{\varepsilon}(T,u)=0,$ each term should be equal with 0. This implies that in the second term of $J_{\varepsilon},$ $P(y(T=0))-Py^{tar}=0$ which is a contradiction with the fact that $Py_{0}\neq Py^{tar}.$ We conclude that the optimal $T_{\varepsilon}^{\ast}$ must be positive.

We consider a minimizing sequence $(T_{\varepsilon}^{n},u_{\varepsilon}^{n})$ with $T_{\varepsilon}^{n}>0$ and $\left\|u_{\varepsilon}^{n}(t)\right\|_{U}\leq\rho,$ satisfying

[TABLE]

Hence, $T_{\varepsilon}^{n}\rightarrow T_{\varepsilon}^{\ast},$ as $n\rightarrow\infty.$ Then, for any $\delta>0$ there exists $n_{\delta}$ such that $T_{\varepsilon}^{n}\geq T_{\varepsilon}^{\ast}-\delta,$ with $\delta$ arbitrarily small, for $n\geq n_{\delta}.$ On a subsequence

[TABLE]

for all $T>0.$ Then, the state system corresponding to any $T>d_{\varepsilon}+1$ and $u_{\varepsilon}^{n}$ has a unique continuous solution satisfying (3) on $(0,T),$ and it tends, as $n\rightarrow\infty,$ to the solution corresponding to $(T,u_{\varepsilon}^{\ast}).$ In particular, this happens for $T=T_{\varepsilon}^{\ast}-\delta$, with $\delta$ arbitrary small. Then, on a subsequence denoted still by $n,$ we have

[TABLE]

[TABLE]

[TABLE]

Next, we proceed in a similar way as in Theorem 3.3 to show that $y^{T_{\varepsilon}^{n},u_{\varepsilon}^{n}}\rightarrow y^{T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast}}:=y_{\varepsilon}^{\ast}$ in the corresponding spaces and that

[TABLE]

These imply that $y_{\varepsilon}^{\ast}$ is the solution to the state system corresponding to $(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast}).$

Let us denote

[TABLE]

Taking $\psi\in U^{\ast}$ we have

[TABLE]

So, we get that $h_{\varepsilon}^{n}(t)\rightarrow h_{\varepsilon}^{\ast}(t)$ weakly in $U,$ for all $t\geq 0$ and

[TABLE]

Passing to the limit in (4.2) we get on the basis of the previous convergences and of the weakly lower semicontinuity of the norms, that $J_{\varepsilon}(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast})=d_{\varepsilon},$ that is $(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast})$ is an optimal controller in $(P_{\varepsilon}).$ $\square$

**Theorem 4.2. **Assume $y_{0}\in V,$ $Py^{tar}\in P(H),$ $Py_{0}\neq Py^{tar}$. Let $(T^{\ast},u^{\ast},y^{\ast})$ be optimal in $(\mathcal{P})$ and $(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast},y_{\varepsilon}^{\ast})$ be optimal in $(\mathcal{P}_{\varepsilon}).$ Then,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

**Proof. *Let $(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast},y_{\varepsilon}^{\ast})$ *be optimal in $(P_{\varepsilon}).$ Then,

[TABLE]

for any $T>0$ and $u\in L^{\infty}(0,\infty;U),$ $\left\|u(t)\right\|_{U}\leq\rho$ a.e. $t>0,$ where $y_{\varepsilon}^{\ast}$ is the solution to the state system corresponding to $(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast})$ and $y$ is the solution to the state system corresponding to $(T,u).$ Let us set in (4), $T=T^{\ast}$ and $u=u^{\ast},$ an optimal controller in $(P).$ Thus, the second and the last term on the right-hand side of (4) are zero and

[TABLE]

Then, $T_{\varepsilon}^{\ast}\rightarrow T^{\ast\ast}$ and $T_{\varepsilon}^{\ast}\geq T^{\ast\ast}-\delta,$ with $\delta$ arbitrarily small, and selecting a subsequence indicated still by $\varepsilon,$ we have

[TABLE]

The solution $y_{\varepsilon}^{\ast}$ satisfies the estimates

[TABLE]

where $X_{T}:=C([0,T],H)\cap L^{\infty}(0,T;V)\cap W^{1,2}(0,T;H)$, and $y_{\varepsilon}^{\ast}\rightarrow y^{\ast\ast}$ in the spaces $X_{T^{\ast\ast}-\delta}$ defined on $(0,T^{\ast\ast}-\delta),$ for $\delta$ arbitrary. As in the previous proof we show that all the convergences (4.3)-(4.5) take place also in the spaces defined on $(0,T^{\ast\ast}).$ Also, we have

[TABLE]

implying that $\left\|y_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})-y^{\ast\ast}(T^{\ast\ast})\right\|_{H}\rightarrow 0,$ strongly in $H,$ as $\varepsilon\rightarrow 0.$ By (4.13) we have

[TABLE]

and so $Py_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})\rightarrow Py^{tar}$ strongly in $H,$ implying the relation $Py^{\ast\ast}(T^{\ast\ast})=Py^{tar}.$ Again by (4.13),

[TABLE]

whence we get at limit that $T^{\ast\ast}\leq T^{\ast}.$ Now $T^{\ast\ast}$ and $u^{\ast\ast}$ satisfy the restrictions required in problem $(P),$ that is $T^{\ast\ast}>0,$ $\left\|u^{\ast\ast}(t)\right\|_{U}\leq\rho,$ and $Py^{\ast\ast}(T^{\ast\ast})=Py^{tar},$ and since $T^{\ast}$ is the infimum in $(P)$ it follows that $T^{\ast\ast}=T^{\ast}.$ Recalling (4.7) we define

[TABLE]

We have by (4.13) that

[TABLE]

and so $T^{\ast\ast}+\limsup\limits_{\varepsilon\rightarrow 0}E_{\varepsilon}(T_{\varepsilon}^{\ast})\leq T^{\ast},$ hence

[TABLE]

Therefore,

[TABLE]

On the other hand, we know that $u_{\varepsilon}^{\ast}\rightarrow u^{\ast\ast}$ weakly in $L^{2}(0,T^{\ast};U),$ so that

[TABLE]

implying that $u^{\ast\ast}=u^{\ast}$ on $(0,T^{\ast}).$

For a later use we prove that

[TABLE]

We write

[TABLE]

Then,

[TABLE]

Let us take $t\in[T_{\varepsilon}^{\ast}-\tau,T_{\varepsilon}^{\ast}]$ with $\tau>\varepsilon,$ and integrate the previous inequality along with $t$ in this interval. We have

[TABLE]

Let us make $\varepsilon$ goes to zero and get by (4.15) that

[TABLE]

which yields (4.17), since $\tau$ is arbitrary. On the basis of (4.13) we write that

[TABLE]

whence

[TABLE]

We conclude that $\lim\limits_{\varepsilon\rightarrow 0}J_{\varepsilon}(T_{\varepsilon}^{\ast\ast},u_{\varepsilon}^{\ast\ast})=T^{\ast\ast}=J(T^{\ast\ast},u^{\ast\ast})$ and so $(T^{\ast\ast},u^{\ast\ast})$ is optimal in $(P)$. But, $T^{\ast}$ is also optimal and unique and it follows $T^{\ast\ast}=T^{\ast}$ and $u^{\ast\ast}=u^{\ast}$ a.e. on $(0,T^{\ast}).$ Eventually, we also have obtained (4)-(4.11), as claimed. $\square$

5 The maximum principle

In this section, besides $(a_{1}),$ $(a_{2}),$ $(b_{1}),$ $(c_{1}),$ we assume $(a_{3})-(a_{6}).$

5.1 The system of first order variations and the dual system

Let us introduce the Cauchy problem

[TABLE]

where $v\in L^{\infty}(0,\infty;U),$ $\left\|v(t)\right\|_{U}\leq C,$ a.e. $t\geq 0.$

**Proposition 5.1. **Problem (5.1) has a unique solution

[TABLE]

Proof. We recall that $A^{\prime}(y_{\varepsilon}^{\ast}(t))$ is continuous from $V$ to $V^{\ast},$ for all $t\geq 0,$ and has the properties (2.17) and (2.5), $\left\|A^{\prime}(y_{\varepsilon}^{\ast}(t))z\right\|_{V^{\ast}}\leq C\left\|z\right\|_{V},$ due to $\left\|y_{\varepsilon}^{\ast}(t)\right\|_{V}\leq C_{T},$ by (3). Then, the result claimed in the statement is ensured by the Lions theorem. $\square$

Let $(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast})$ be an optimal controller in $(\mathcal{P}_{\varepsilon}).$ For $\lambda>0,$ we set

[TABLE]

In this way we can give variations to both controllers, if $P=I,$ or to the first component in the case $P\neq I.$ We define $Y_{\lambda}=\frac{y_{\varepsilon}^{\lambda}-y_{\varepsilon}^{\ast}}{\lambda},$ where $y_{\varepsilon}^{\lambda}$ is the solution to the state system (1.1)-(1.2) corresponding to $u_{\varepsilon}^{\lambda}$ and $T_{\varepsilon}^{\ast}.$

**Proposition 5.2. **Let $Y$ be the solution to (5.1) and let $T>0$. We have

[TABLE]

which ensures that (5.1) is just the system of first order variations related to (1.1)-(1.2).

**Proof. **Let us define

[TABLE]

We write the equation for $y_{\varepsilon}^{\lambda}$ subtract the equation for $y_{\varepsilon}^{\ast},$ divide by $\lambda$ and subtract the equation (5.1). The equation verified by $\zeta_{\lambda}$ reads

[TABLE]

Now, we can represent the third term as

[TABLE]

and so, the equation becomes

[TABLE]

We test (5.7) by $\zeta_{\lambda}(t),$ integrate with respect to $t,$ and get, by (2.17) and (2.5) that

[TABLE]

By Gronwall’s lemma we obtain

[TABLE]

We recall that by (3), we have $\left\|y_{\varepsilon}^{\ast}(t)\right\|_{V}\leq e^{CT}\left(\left\|y_{0}\right\|_{V}^{2}+\rho^{2}T\right)\leq C_{T},$ for all $t\in[0,T],$ and

[TABLE]

Here, $C_{T}$ may change from line to line. Moreover, by (2.19)

[TABLE]

[TABLE]

We also recall (3.3) which yields, for all $t\in[0,T_{\varepsilon}^{\ast}],$ that

[TABLE]

and so

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

for $\nu$ fixed, implying that

[TABLE]

This yields that

[TABLE]

by $(a_{4})$ and (7.3). We denote $f_{\lambda}(t):=\left\|A^{\prime}((\nu y_{\varepsilon}^{\lambda}+(1-\nu)y_{\varepsilon}^{\ast})(t))-A^{\prime}(y_{\varepsilon}^{\ast}(t))\right\|_{L(V,V^{\ast})}^{2}$ and infer that $f_{\lambda}(t)\rightarrow 0$ a.e. $t\in(0,T),$ and that $\left|f_{\lambda}(t)\right|\leq C.$ This implies, by the Lebesgue dominated convergence theorem, that $f_{\lambda}\rightarrow 0$ in $L^{2}(0,T).$ Thus, by (5.1)

[TABLE]

This proves (5.4). $\square$

Now, we introduce the adjoint system

[TABLE]

[TABLE]

**Proposition 5.3. **Let $Py^{tar}\in P(D_{H})$ and assume (2.8). Then, for each $\varepsilon>0,$ problem (5.13)-(5.14) has a unique solution

[TABLE]

**Proof. **By Theorem 3.2 we deduce that $p_{\varepsilon}(T_{\varepsilon}^{\ast})\in D_{H},$ since $P(y_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})-y^{tar})\in P(D_{H}).$ If $P\neq I,$ the second component of $p_{\varepsilon}$ is $0\in D_{H}.$ We use the transformation $t\rightarrow T_{\varepsilon}^{\ast}-t$ and so (5.13)-(5.14) transforms into a forward equation. The operator $(A^{\prime}(y_{\varepsilon}^{\ast}(t)))^{\ast}$ is continuous from $V$ to $V^{\ast}$ for all $t\geq 0$ and satisfies the properties of Lions theorem, such that we can deduce, as in Proposition 5.1, that (5.13)-(5.14) has a unique solution

[TABLE]

A first estimate is obtained by testing (5.13) by $p_{\varepsilon}(t)$ and integrating over $(t,T_{\varepsilon}^{\ast}).$ Using (2.17), this yields

[TABLE]

To prove the additional regularity we multiply (5.13) by $\Gamma_{\nu}p_{\varepsilon}(t),$ integrate over $(t,T_{\varepsilon}^{\ast})$ and use (2.8). We have

[TABLE]

Taking into account (5.17) and (3), that is $\left\|y_{\varepsilon}^{\ast}(\tau)\right\|_{V}\leq C_{T_{\varepsilon}^{\ast}}<C_{T^{\ast}+1},$ we obtain

[TABLE]

for all $t\in[0,T_{\varepsilon}^{\ast}],$ since $\left\|\Gamma_{H}p_{\varepsilon}(\tau)\right\|_{H}=\left\|p_{\varepsilon}(\tau)\right\|_{D_{H}}.$

Here, we used the relation $\left\|\Gamma_{\nu}p_{\varepsilon}(T_{\varepsilon}^{\ast})\right\|_{H}\leq\left\|\Gamma_{H}p_{\varepsilon}(T_{\varepsilon}^{\ast})\right\|_{H}$ for $p_{\varepsilon}(T_{\varepsilon}^{\ast})\in D_{H}.$ Now, we can pass to the limit as $\nu\rightarrow 0$ and obtain

[TABLE]

and so by (5.1) we get

[TABLE]

Thus, $p_{\varepsilon}\in L^{2}(0,T_{\varepsilon}^{\ast};D_{H})\cap L^{\infty}(0,T_{\varepsilon}^{\ast};V).$ For a.a. $t\in(0,T_{\varepsilon}^{\ast})$ we still have

[TABLE]

By (5.13) it follows that $p_{\varepsilon}^{\prime}\in L^{2}(0,T_{\varepsilon}^{\ast};H)$ and so (5.15) is proved. $\square$

5.2 Approximating optimality conditions

Let us introduce the sets

[TABLE]

and denote the normal cone to $K$ at $w$ by

[TABLE]

and the normal cone to $\mathcal{K}_{T}$ at $\omega$ by

[TABLE]

We recall (see e.g., [4]) that $\chi\in N_{\mathcal{K}_{T}}(\omega)$ iff $\chi(t)\in N_{K}(\omega(t))$ a.e. $t\in(0,T).$

We denote by $F:U\rightarrow U^{\ast}$ the duality mapping of $U$ (see (7.4) in the Appendix) and recall that $h_{\varepsilon}^{\ast}(t)$ was defined in (4.7), $h_{\varepsilon}^{\ast}(t)=\int_{0}^{t}P(u_{\varepsilon}^{\ast}-u^{\ast})(\tau)d\tau.$

**Proposition 5.4. **Assume

[TABLE]

Let $(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast})$ be an optimal control in $(P_{\varepsilon})$ with the optimal state $y_{\varepsilon}^{\ast}$. Then,

[TABLE]

*and *

[TABLE]

where $p_{\varepsilon}$ *is the solution to the adjoint equation *(5.13)-(5.14). Moreover, $t\rightarrow u_{\varepsilon}^{\ast}(t)$ turns out to be continuous on $[0,T_{\varepsilon}^{\ast}].$

**Proof. **Let $(T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast})$ be an optimal controller in $(P_{\varepsilon}).$ We shall compute separate variations with respect to $T_{\varepsilon}^{\ast}$ and $u_{\varepsilon}^{\ast}$. By the condition of optimality for $u_{\varepsilon}^{\ast}$ we have

[TABLE]

In particular, replacing $u$ by $u_{\varepsilon}^{\lambda}=Pu_{\varepsilon}^{\ast}+\lambda v$ with $v=P(\overline{u}-u_{\varepsilon}^{\ast}),$ $\overline{u}(t)\in K,$ and performing some calculations, recalling (7.11) we get

[TABLE]

Observing that

[TABLE]

we obtain

[TABLE]

Here we used that $P^{2}=P$ and the fact that $(Pw,P\overline{w})_{H}$ is the same with $(Pw,\overline{w})$ when $Pw=(w_{1},0).$ We test (5.1) by $p_{\varepsilon}(t)$ and integrate over $(0,T_{\varepsilon}^{\ast}).$ By a straightforward calculation we obtain

[TABLE]

Using again the adjoint system, this equation reduces to

[TABLE]

We recall that $v=P(\overline{u}-u_{\varepsilon}^{\ast}).$ Replacing the left-hand side of (5.27) into (5.26) we deduce that

[TABLE]

for all $\overline{u}(t)\in K,$ that is $\overline{u}(t)\in U,$ $\left\|\overline{u}(t)\right\|_{U}\leq\rho$ a.e. $t\geq 0.$ This yields

[TABLE]

for all $\overline{u}(t)\in K,$ a.e. $t\in(0,T_{\varepsilon}^{\ast}),$ and implies, by (5.19), that

[TABLE]

or, equivalently,

[TABLE]

Moreover, relation (5.30) implies that $t\rightarrow u_{\varepsilon}^{\ast}(t)$ is continuous, because $(\varepsilon F+N_{K})^{-1}$ is single-valued and Lipschitz continuous, the integral is continuous and $p_{\varepsilon}$ belongs to $C([0,T_{\varepsilon}^{\ast}];H),$ so that (5.30) is true for all $t\in[0,T_{\varepsilon}^{\ast}].$ We also note that

[TABLE]

We recall by (4.14) that $A_{H}y_{\varepsilon}^{\ast}(t)\in H,$ $(y_{\varepsilon}^{\ast})^{\prime}(t)\in H$ a.e. $t\in(0,T_{\varepsilon}^{\ast}).$ Also, $y_{\varepsilon}^{\ast}\in C_{w}([0,T_{\varepsilon}^{\ast}];V)$ and $Ay\in C_{w}([0,T_{\varepsilon}^{\ast}];V^{\ast})$ (see the observation after Theorem 3.2). By the state equation we have $(y_{\varepsilon}^{\ast})^{\prime}=-Ay_{\varepsilon}^{\ast}+Bu_{\varepsilon}^{\ast}\in C_{w}([0,T_{\varepsilon}^{\ast}];V^{\ast}).$ Indeed, $y_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})\in V,$ so $Ay_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})\in V^{\ast}$ and $Bu_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})\in H.$ Thus, $(y_{\varepsilon}^{\ast})^{\prime}(T_{\varepsilon}^{\ast})\in V^{\ast},$ for all $\varepsilon>0.$ Recalling (5.14) and that $Py_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})-Py^{tar}\in P(D_{H}),$ we deduce that $p_{\varepsilon}(T_{\varepsilon}^{\ast})\in D_{H},$ for all $\varepsilon>0.$

Next, we keep $u_{\varepsilon}^{\ast}$ fixed and give variations to $T_{\varepsilon}^{\ast}$. Since $T_{\varepsilon}^{\ast}$ realizes the minimum in $(P_{\varepsilon})$ we can write

[TABLE]

that is,

[TABLE]

In these calculations we took into account that $u_{\varepsilon}^{\ast}$ and the solution to the approximating state are continuous with respect to $t\in[0,\infty).$ Then, the solution $y_{\varepsilon}^{T_{\varepsilon}^{\ast}+\lambda,u_{\varepsilon}^{\ast}}(t)$ calculated for $t\in(0,T_{\varepsilon}^{\ast}+\lambda)$ and $u_{\varepsilon}^{\ast},$ restricted to $(0,T_{\varepsilon}^{\ast})$ coincides with $y_{\varepsilon}^{T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast}}(t)$ the solution calculated on $(0,T_{\varepsilon}^{\ast})$, which was denoted by $y_{\varepsilon}^{\ast}(t).$ Performing some calculations we get

[TABLE]

Doing the same for $T_{\varepsilon}^{\ast}-\lambda$ and observing that the solution $y_{\varepsilon}^{T_{\varepsilon}^{\ast},u_{\varepsilon}^{\ast}}(t),$ calculated for $t\in(0,T_{\varepsilon}^{\ast})$ and $u_{\varepsilon}^{\ast},$ restricted to $(0,T_{\varepsilon}^{\ast}-\lambda)$ is in fact $y_{\varepsilon}^{T_{\varepsilon}^{\ast}-\lambda,u_{\varepsilon}^{\ast}}(t)$ the solution calculated on $(0,T_{\varepsilon}^{\ast}-\lambda),$ we get the reverse inequality. Finally, we obtain

[TABLE]

Then, using the state system (1.1) for $y_{\varepsilon}^{\ast}$ and the final conditions of the adjoint system, we can express the term

[TABLE]

Plugging this in (5.31), we obtain

[TABLE]

We replace $Pu_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})$ from (5.30),

[TABLE]

which can be still written

[TABLE]

By using this and (7.9) we obtain for the second term in (5.32)

[TABLE]

Therefore, (5.32) becomes

[TABLE]

which finally can be written

[TABLE]

The next calculation can be performed due to the supplementary regularity of $p_{\varepsilon},$ that is $p_{\varepsilon}^{\prime}\in L^{2}(0,T_{\varepsilon}^{\ast};H),$ given by (5.15). We multiply scalarly the state equation by $p_{\varepsilon}^{\prime}(t)$, add with the adjoint equation multiplied by $(y_{\varepsilon}^{\ast})^{\prime}(t)$, getting

[TABLE]

a.e. $t\in(0,T_{\varepsilon}^{\ast}),$ that reduces to

[TABLE]

We integrate on $(t,T_{\varepsilon}^{\ast})$ and obtain

[TABLE]

since $\left\langle u_{\varepsilon}^{\ast}(s),Pk(s)\right\rangle_{U,U^{\ast}}=\left\langle Pu_{\varepsilon}^{\ast}(s),Pk(s)\right\rangle_{U,U^{\ast}}$where $k(s)$ is either $p_{\varepsilon}(s)$ or $\int_{s}^{T_{\varepsilon}^{\ast}}F(h_{\varepsilon}^{\ast}(\tau))d\tau$, both containing $P$ in their expressions. Denoting

[TABLE]

we see by (5.30) that $\zeta_{\varepsilon}(t)=(\varepsilon F+N_{K})(Pu_{\varepsilon}^{\ast}(t)).$ By (7.14),

[TABLE]

Thus, we can express (5.30) as

[TABLE]

(see (7.10)-(7.14)). Then, the integrand of the first term on the right-hand side in (5.2) becomes

[TABLE]

Plugging this in (5.2) we get

[TABLE]

for all $t\in[0,T_{\varepsilon}^{\ast}].$ By comparison with (5.34), we obtain

[TABLE]

Recalling (5.2), this yields

[TABLE]

and so we obtain (5.23), as claimed. $\square$

5.3 Optimality conditions for $(P)$

In order to ensure the passing to the limit in the approximating optimality conditions (5.22)-(5.23) we complete the hypotheses $(a_{1})-(a_{6}),$ $(b_{1}),$ $(c_{1})$ with $(d_{1})-(d_{5})$.

**Theorem 5.5. **Let

[TABLE]

Let $(T^{\ast},u^{\ast},y^{\ast})$ be an optimal pair in $(P).$ Then, the first order necessary conditions of optimality are

[TABLE]

[TABLE]

where $y^{\ast}$ is the solution to the state system (1.1)-(1.2) corresponding to $(T^{\ast},u^{\ast}),$ and $p$ *is a solution to *

[TABLE]

[TABLE]

**Proof. **First, we prove that

[TABLE]

with** $C$** independent of $\varepsilon.$

Let us begin with the case $P(y_{1},y_{2})=(y_{1},0),$ $B(u_{1},u_{2})=(u_{1},0).$ We recall that in this case $y^{tar}=(y_{1}^{\mbox{target}},z),$ $\forall z\in H,$ and $Py^{tar}=(y_{1}^{\mbox{target}},0).$ We start from (5.34) and express the third term on the left-hand side as

[TABLE]

where $\widehat{y}=(y_{1}^{\mbox{target}},\widehat{z})$ set by (2.13), satisfying (2.14), with the choice (2.15). We note that

[TABLE]

Then, by (5.34), we can write

[TABLE]

Here, we took into account that

[TABLE]

Now, we use (2.14) which is assumed to take place for $t\in(0,T_{\ast}+\delta)$, with $T_{\ast}$ the time specified in the controllability hypothesis $(c_{1}),$ with $T_{\ast}\geq T^{\ast}.$ Recall that $T_{\varepsilon}^{\ast}\rightarrow T^{\ast}.$ Hence, for $\varepsilon$ sufficiently small, $T_{\varepsilon}^{\ast}\in(0,T^{\ast}+\delta)\subset(0,T_{\ast}+\delta),$ with $\delta$ arbitrary small and it follows that relation (2.14) can take place also for $t=T_{\varepsilon}^{\ast}$, that is

[TABLE]

Here we used (5.45). Then,

[TABLE]

where we took into account (2.12). We recall (4.18),

[TABLE]

and so, by (2.15), we can write

[TABLE]

because $\left\|h_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})\right\|_{U}^{2}+\varepsilon C\rho^{2}\rightarrow 0.$ The convergence of the first term is due to (4.17). This yields

[TABLE]

and choosing $\rho>\rho_{1}$ we finally get (5.44).

As a matter of fact, in the proof of (5.44) the second component $\widehat{z}$ of $\widehat{y}$ can be generally set as the second component of the approximating state solution.

If $P=I,$ we proceed in the same way, and use that $y^{tar}=(y_{1}^{\mbox{target}},y_{2}^{\mbox{target}})$ and (d_{5})\and take $Ay^{tar}$ instead of $A\widehat{y}.$ We have

[TABLE]

which tends to zero by (4.18). Here, $Py_{\varepsilon}(T_{\varepsilon}^{\ast})-Py^{tar}$ has both nonzero components.

We recall the adjoint system given by (5.13)-(5.14). Since the final data is bounded in $U^{\ast},$ according to (5.44), we expect to obtain at limit a solution with a weaker regularity. We are going to obtain some uniform estimates for the solution $p_{\varepsilon}.$

A first estimate is obtained by multiplying scalarly (5.13) by $\Gamma_{H}^{-1}p_{\varepsilon}(t)$ and integrating from $t$ to $T_{\varepsilon}^{\ast}$

[TABLE]

where the first term on the right-hand side was obtained by using the properties of the duality mapping (7.15). According to (2.11) and (2.12) for $v=p_{\varepsilon}(T_{\varepsilon}^{\ast})\in V^{\ast},$ we successively get

[TABLE]

Here, we used (3). By Gronwall lemma and (5.44) we obtain

[TABLE]

independently on $\varepsilon.$

Next, we multiply scalarly (5.13) by $\Gamma_{H}^{-\alpha}p_{\varepsilon}(t)$ (where $\alpha$ is chosen by (2.9) and (2.10)) and integrate from $t$ to $T_{\varepsilon}^{\ast}.$ Applying (2.10) and (2.9) we obtain

[TABLE]

Recalling (5.48) we obtain $\left\|\Gamma_{H}^{(1-\alpha)/2}p_{\varepsilon}\right\|_{L^{2}(0,T_{\varepsilon}^{\ast};H)}\leq C,$ that is

[TABLE]

To use further these estimate we have to modify the functional framework in the following sense. We extend the operator $(A_{H}^{\prime}(y_{\varepsilon}^{\ast}(t)))^{\ast}$ to $H,$ for all $t\in[0,T_{\varepsilon}^{\ast}],$ namely we define $\tilde{A}_{H}^{\prime}(y_{\varepsilon}^{\ast}(t)):H\subset D_{H}^{\ast}\rightarrow D_{H}^{\ast}$ by

[TABLE]

The norm on $D_{H}^{\ast}$ is defined by $\left\|\theta\right\|_{D_{H}^{\ast}}=\left\|(A_{H}^{\prime}(y_{\varepsilon}^{\ast}(t)))^{-1}\theta\right\|_{H}.$ We have, by (2.6)

[TABLE]

which yields for $v=p_{\varepsilon}$,

[TABLE]

since $\left\|y_{\varepsilon}^{\ast}(t)\right\|_{V}\leq C.$ By comparison in the adjoint equation (5.13) we obtain

[TABLE]

We recall that $T_{\varepsilon}^{\ast}\rightarrow T^{\ast}$ and so $T^{\ast}-\delta\leq T_{\varepsilon}^{\ast},$ with $\delta$ arbitrary, so that the estimates are true also on $(0,T^{\ast}-\delta).$ By (5.48) and the latter, selecting a subsequence, denoted still by $\varepsilon,$ we have

[TABLE]

[TABLE]

Because $\delta$ is arbitrary, the previous convergences take place on $\cap_{\delta>0}(0,T^{\ast}-\delta)=(0,T^{\ast})$.

Since $D\left(\Gamma_{H}^{(1-\alpha)/2}\right)$ is compact in $H$ and $H\subset D_{H}^{\ast},$ we have by Aubin-Lions lemma that

[TABLE]

Then, using the convergence $y_{\varepsilon}^{\ast}(t)\rightarrow y^{\ast}(t)$ strongly in $V$ a.e. $t,$ and the continuity (2.7), we have

[TABLE]

for $\psi\in L^{2}(0,T_{\varepsilon}^{\ast};D_{H})$, which implies by the previous convergences that

[TABLE]

We also have

[TABLE]

By these convergences we obtain (5.42) in the sense of distributions and a.e.

We go back now to (5.22), and recall (5.29) which can be equivalently written

[TABLE]

We pass to the limit as $\varepsilon\rightarrow 0$ and have

[TABLE]

[TABLE]

by (4.16), and

[TABLE]

Therefore,

[TABLE]

But $N_{\mathcal{K}_{T^{\ast}}}$ is maximal monotone from $L^{2}(0,T^{\ast};U)$ to $L^{2}(0,T^{\ast};U^{\ast})$, that is weakly-strongly closed and since $Pu_{\varepsilon}^{\ast}\rightarrow Pu^{\ast}$ weakly in $L^{2}(0,T^{\ast};U),$ we get $-B^{\ast}p\in N_{\mathcal{K}_{T^{\ast}}}(Pu^{\ast}),$ or equivalently (5.40).

Finally, we have to pass to the limit in (5.23). For this, we integrate (5.23) from $s$ to $s^{\prime},$ $0<s<s^{\prime}<T^{\ast}$ and get

[TABLE]

We recall that $Ay_{\varepsilon}^{\ast}\rightarrow Ay^{\ast}$ weakly in $L^{2}(0,T^{\ast};H)$ and note that

[TABLE]

Finally, $\left\|h_{\varepsilon}^{\ast}(T_{\varepsilon}^{\ast})\right\|_{U}^{2}\rightarrow 0,$ by (4.17). We pass to the limit as $\varepsilon$ goes to 0 and get

[TABLE]

Dividing by $(s^{\prime}-s)$ and passing to the limit as $s\rightarrow s^{\prime}$ we obtain (5.41), for a.e. $t\in(0,T^{\ast})$. $\square$

In the case when $U=U^{\ast}=H$ we have a particular result for which we assume the hypotheses $(a_{1})-(a_{6})$ and replace $(d_{1})-(d_{5})$ by simpler ones.

**Corollary 5.6. **Let $U=U^{\ast}=H$ and assume (5.39), (2.14), and

[TABLE]

[TABLE]

[TABLE]

(instead of (2.12)). Then, (5.40)-(5.42) take place and $p(T^{\ast})\in H.$

**Proof. **We resume the proof of the estimate for $\left\|B^{\ast}p_{\varepsilon}(T_{\varepsilon}^{\ast})\right\|_{U^{\ast}}$ in Theorem 5.5 and have now in (5.3)

[TABLE]

Since $U^{\ast}=H$ we get $\left\|B^{\ast}p_{\varepsilon}(T_{\varepsilon}^{\ast})\right\|_{H}\leq C,$ which will ensure a more regular solution for $p.$ We multiply (5.13) by $p_{\varepsilon}(t),$ integrate from $t$ to $T_{\varepsilon}^{\ast}$ and use (5.54) to obtain

[TABLE]

Then,

[TABLE]

and by (5.13) we infer that

[TABLE]

On a subsequence we obtain

[TABLE]

where $p$ turns out to be the solution to (5.42). The rest of the proof can be led as in Theorem 5.5. $\square$

Remark 5.7. Consider the case $P=I$ and $y^{tar}=0$ and assume that, for each $\widetilde{y}\in C([0,T];H)\cap L^{2}(0,T;V)$, the linearized problem

[TABLE]

is exactly null controllable in the following sense: for each $Y_{0}\in H$ with $\left\|Y_{0}\right\|_{H}\leq 1,$ there is $v\in L^{2}(0,T;U),$ with $\left\|v\right\|_{L^{2}(0,T;U)}\leq\gamma,$ such that $Y(T)=0.$ Then, Theorem 5.5 remains true, without assuming $(d_{1})-(d_{5}).$ Here there is the argument. By the above controllability hypothesis, we get for the dual equation, $-p^{\prime}(t)+(A^{\prime}(\widetilde{y}))^{\ast}p(t)=0,$ the following observability inequality:

[TABLE]

and therefore

[TABLE]

Then, substituting in (5.23) we get $\left\|p_{\varepsilon}(t)\right\|_{H}\leq\gamma_{1}\int_{t}^{T_{\varepsilon}^{\ast}}\left\|B^{\ast}p_{\varepsilon}(\tau)\right\|_{U}^{2}d\tau+C(\varepsilon)+C\leq C_{t},$ for all $t\in[0,T_{\varepsilon}^{\ast}]$ and then $\int_{0}^{T_{\varepsilon}^{\ast}-\delta}\left\|p_{\varepsilon}(t)\right\|_{H}^{2}dt\leq C_{\delta},$ $\delta>0.$ Thus, we may pass to the limit in (5.13)-(5.14) to get (5.40)-(5.42).

6 Examples

We particularize our results to some equations and systems modelling various processes in physical applications. Let $\Omega$ be an open bounded subset of $\mathbb{R}^{d},$ $d\leq 3$, with a sufficient regular boundary $\partial\Omega$ and let $\nu$ be the outward normal to $\partial\Omega.$ Let $L^{r}(\Omega)$ be the space of $r$-summable functions, $y:\Omega\rightarrow\mathbb{R},$ with the norm $\left\|y\right\|_{r}=\left(\int_{\Omega}\left|y\right|^{r}dx\right)^{1/r}$, $1\leq r<\infty,$ and $\left\|y\right\|_{\infty}=ess\sup_{x\in\Omega}\left|y(x)\right|$ for $r=\infty.$ The spaces $H^{r}(\Omega):=W^{1,r}(\Omega),$ with $1\leq r<\infty,$ and $H_{0}^{1}(\Omega)$ are the standard Sobolev spaces, and $H^{-1}(\Omega)$ is the dual of $H_{0}^{1}(\Omega).$

Example 1. Diffusion equation with a potential and drift term.

Let us consider the problem

[TABLE]

where

[TABLE]

[TABLE]

This problem characterizes the evolution of a diffusion process under the influence of a potential $\beta$ and of a drift term $\nabla\cdot(by).$ For $\beta=0$ the model can describe the diffusion with transport of a substance in a fluid. If $b=0,$ $\beta(y)=y^{3}$ and $a_{1}=-1$ we note that this is the Allen-Cahn equation describing the phase transitions of a material, which can exists in different phases, under the influence of a double-well potential. Such a problem with different assumptions for $\beta$ was treated in [3], Section 6.1.4.

We study problem $(\mathcal{P})$ for $u\in L^{\infty}(0,\infty;L^{2}(\Omega))$ with $\left\|u(t)\right\|_{H}\leq\rho$ a.e. $t\geq 0.$

Proposition 6.1. Let $y_{0}\in V,$ $y^{\mbox{target}}\in D_{H},$ $\mathit{d=}-\Delta y^{\mbox{target}}+\beta(y^{\mbox{target}})+a_{1}y^{\mbox{target}}-\nabla\cdot(by^{\mbox{target}})\in L^{2}(\Omega)$ and $\rho>\left\|d\right\|_{L^{2}(\Omega)}.$* Then, there exists $(T^{\ast},u^{\ast})$ solution to $(\mathcal{P})$* *satisfying *(5.40)-(5.41), where $p$ solves

[TABLE]

**Proof. **Let us set:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We shall check first the hypotheses $(a_{1})-(c_{1})$ in Section 3. Since $\beta$ is maximal monotone we have

[TABLE]

implying that $\lambda I+A$ is coercive for $\lambda$ large. Here, we used the trace theorem, $\left\|y\right\|_{L^{2}(\partial\Omega)}\leq C_{tr}\left\|y\right\|_{V},$ with $C_{tr}$ is a constant.

By (6.2) it follows that $\left|\beta(r)\right|\leq C\left|r\right|^{\kappa+1}+\left|r\right|,$ and so, for $\kappa\in[0,2]$ we have that

[TABLE]

Let $y_{n}\rightarrow y$ strongly in $V.$ Since $\left\|\beta(y_{n})\right\|_{H}\leq C$ it follows that $\beta(y_{n})\rightarrow\beta(y)$ weakly in $H$ because $\beta$ is strongly-weakly closed. Moreover, we have $\beta(y_{n})\rightarrow\beta(y)$ a.e. on $\Omega,$ and so $\beta(y_{n})\rightarrow\beta(y)$ strongly in $H,$ by Vitali’s theorem. Therefore, it follows that $A$ is continuous from $V$ to $V^{\ast}.$ Then,

[TABLE]

hence $\left\|Ay\right\|_{V^{\ast}}$ is bounded on bounded subsets.

Relation (2.4) is immediately verified, because

[TABLE]

since $(\beta(y),-\Delta y)_{H}\geq 0$ by the monotonicity of $\beta.$

The controllability $(c_{1})$ follows by Proposition 7.1.

Next we verify $(a_{3})-(a_{5})$ in Section 5.1. We introduce $A^{\prime}(y):V\rightarrow V^{\ast},$

[TABLE]

then $A_{H}^{\prime}(y):D_{H}\rightarrow H,$ $A_{H}^{\prime}(y)z=-\Delta z+\beta^{\prime}(y)z+a_{1}z-\nabla\cdot(bz),$ and

[TABLE]

and provide first some estimates. Using the Hölder inequality we have

[TABLE]

where $1/q+1/q^{\prime}=1.$ Now, we recall the embedding $W^{s,m}(\Omega)\subset L^{r}(\Omega),$ where $d>sm,$ $m\leq r\leq\frac{dm}{d-sm}$ (see [1], p. 217, Theorem 7.57) and apply it for $m=2,$ $r=2q^{\prime},$ $s=1-\alpha,$ for $\alpha\in(0,1)$ to get

[TABLE]

with $q^{\prime}>1.$ Then, $\left\|y\right\|_{2\kappa q}\leq C\left\|y\right\|_{V}$ if $2\kappa q\leq 6.$ Thus, we obtain

[TABLE]

To this end we must have $3>2(1-\alpha)$ which is satisfied for $\alpha\in[0,1]$ and

[TABLE]

In particular, these are true for $\kappa\leq 2$, $q^{\prime}\geq 3.$ Then

[TABLE]

[TABLE]

Moreover, $y\rightarrow A^{\prime}(y)z$ is continuous from $V$ to $L(V,V^{\ast})$. Indeed, let $y_{n}\in V,$ $y_{n}\rightarrow y$ strongly in $V.$ Then, as before, $\beta^{\prime}(y_{n})\rightarrow\beta^{\prime}(y)$ strongly in $H.$ Therefore,

[TABLE]

Similarly, let $y_{n},$ $y\in D_{H},$ $y_{n}\rightarrow y$ strongly in $V$ and $z\in D_{H}\subset C(\overline{\Omega}).$ Then,

[TABLE]

To prove hypothesis $(a_{6}),$ equivalently (2.8), we calculate

[TABLE]

Here, we used the last inequality in (6.4) and the following relations

[TABLE]

[TABLE]

[TABLE]

Since $U=H$ it remains to check the hypotheses (5.54),

[TABLE]

and (5.55) which is automatically verified with $C^{\ast},$ for $\rho$ large enough. Thus, Corollary 5.6 can be applied. $\square$

We remark, that in virtue of Remark 5.7, Proposition 6.1 applies to equation (6.1) with an internal controller

[TABLE]

where $\Omega_{0}$ is an open subset of $\Omega$ and $1_{\Omega_{0}}$ is the characteristic function of $\Omega_{0}.$ Indeed, by [15], the corresponding linearized system is exactly null controllable.

Example 2. Porous media equation.

Let us consider the porous media equation

[TABLE]

where

[TABLE]

The hypothesis for $\kappa$ places the equation in the slow diffusion case. We study problem $(\mathcal{P})$ for $u\in L^{\infty}(0,\infty;H^{-1}(\Omega)),$ $\left\|u(t)\right\|_{H^{-1}(\Omega)}\leq\rho,$ a.e. $t\geq 0.$

Proposition 6.2. Let $y^{\mbox{target}}\in H_{0}^{1}(\Omega),$ $\Delta\beta(y^{\mbox{target}})\in H^{-1}(\Omega),$* $y_{0}\in H_{0}^{1}(\Omega),$* $\int_{0}^{y_{0}}\beta(s)ds\in L^{1}(\Omega).$* Then, there exists $T^{\ast},$ $u^{\ast}$* and $y^{\ast}$* solution to $(\mathcal{P})$* *satisfying *(5.40)-(5.41), where $U,H,V$ are chosen below and $p\in C_{w}([0,T^{\ast}];H^{-1}(\Omega))\cap L^{2}(0,T^{\ast};L^{2}(\Omega))$ is the solution to

[TABLE]

**Proof. **The proof is led in three steps. First, we prove an intermediate result for $\beta$ having the properties

[TABLE]

Then, we consider (6.8) by replacing $\beta$ by the Yosida approximation $\beta_{\nu}$ which has the properties (6.11) and obtain the minimum time controllability for the approximating solution $y_{\nu}.$ Third, we pass to the limit as $\nu\rightarrow 0.$ To this end, we choose

[TABLE]

where $(L^{2}(\Omega))^{\ast}$ is the dual of $L^{2}(\Omega)$ in the pairing with $H^{-1}(\Omega)$ as pivot space. Moreover,

[TABLE]

We define the operator $A:V\rightarrow V^{\ast}$ by

[TABLE]

and $A_{H}:D_{H}\subset H\rightarrow H$ by $A_{H}y=-\Delta\beta(y).$

The norm on $V^{\ast}=(L^{2}(\Omega))^{\ast}$ is given by $(\theta,\theta)_{V^{\ast}}=\left\|\psi\right\|_{L^{2}(\Omega)},$ where $\theta=A\psi.$

The controllability $(c_{1})$ follows by Proposition 7.1. We begin to check the hypotheses of Corollary 5.6. First,

[TABLE]

which implies the coercivity, too. Then,

[TABLE]

We have $A_{H}^{\prime}(y)z=-\Delta(\beta^{\prime}(y)z)$ and $(A^{\prime}(y))^{\ast}z=-\Delta(\beta^{\prime}(y)z)$, where $\beta^{\prime}(y)p\in V=L^{2}(\Omega).$ Next,

[TABLE]

and if $y_{n}\rightarrow y$ strongly in $V=L^{2}(\Omega),$ we have

[TABLE]

This follows by Lebesgue dominated convergence theorem since $\beta^{\prime}(y_{n})z\rightarrow\beta^{\prime}(y)z$ a.e. on $\Omega$ and $\left|(\beta^{\prime}(y_{n})-\beta^{\prime}(y))z\right|_{L^{2}(\Omega)}\leq 2M_{1}\left|z\right|.$ Then, since we can write $\beta(r)=a_{0}r+\beta_{1}^{\prime}(r)$ we have

[TABLE]

Finally, we have to check (5.54), that is

[TABLE]

while (5.55) which is automatically verified. Thus, we get a minimum time and a controller satisfying the thesis of Corollary 5.6.

In the second step we replace $\beta$ by $\beta_{\nu}$ in (6.8). Both $\beta_{\nu}^{\prime}$ and $\beta_{\nu}^{\prime\prime}$ are bounded by constants $C_{\nu},$ for each $\nu>0.$ On the basis of the previous result we obtain that there exists $T_{\nu}^{\ast},$ $u_{\nu}^{\ast}$ and $y_{\nu}^{\ast}$ satisfying

[TABLE]

[TABLE]

where $y_{\nu}^{\ast}$ is the solution to the approximating state system (6.8) (with $\beta_{\nu}),$ corresponding to $(T_{\nu}^{\ast},u_{\nu}^{\ast}),$ and $p_{\nu}$ is a solution to* *

[TABLE]

[TABLE]

A first estimate for $y_{\nu}^{\ast}$ reads

[TABLE]

where $C$ denote several constants. By multiplying the approximating equation (6.8) by $\beta_{\nu}(y_{\nu}^{\ast}(t))$ and integrating on $(0,t)$ we obtain

[TABLE]

where $\partial j_{\nu}(r)=\beta_{\nu}(r)$ and $\partial j(r)=\beta(r)$ for all $r\in\mathbb{R}.$ This implies

[TABLE]

Since $j(r)=\int_{0}^{r}\beta(s)ds$ and $j(y_{0})\in L^{1}(\Omega)$ it follows that the right-hand side in (6.17) is bounded independently of $\nu.$ This yields

[TABLE]

Then, we multiply (6.14) by $p_{\nu}(t)$ and integrate over $(t,T_{\nu}^{\ast}),$ getting

[TABLE]

Next, we determine an estimate for $A^{\prime}(y_{\nu}^{\ast})p_{\nu}$ and begin by computing

[TABLE]

where $\frac{1}{q^{\prime}}=1-\frac{q}{2},$ that is $q^{\prime}=\frac{2}{2-q},$ for $1<q<2$ and $\kappa q^{\prime}=\frac{2\kappa q}{2-q}\leq 2,$ meaning that $q\leq\frac{2}{\kappa+1}$, which is true if $\kappa<1.$ Therefore, by (6.16) and (6.19) we obtain

[TABLE]

This implies that

[TABLE]

where $X$ is the image of $L^{q}(\Omega)$ by the operator $-\Delta.$ More precisely, $X$ is the completion of $L^{q}(\Omega)$ in the norm $\left|\left\|w\right\|\right|_{X}=\left\|A^{-1}w\right\|_{L^{q}(\Omega)}.$ Moreover, applying the same argument as in Theorem 5.5 we can deduce that $T_{\nu}^{\ast}\rightarrow T^{\ast}$, and on a subsequence, it follows that

[TABLE]

[TABLE]

since $y_{\nu}\rightarrow y$ strongly, $\beta_{\nu}(y_{\nu}^{\ast})\rightarrow\eta$ weakly in $L^{2}(0,T_{\nu}^{\ast};L^{2}(\Omega))$ and $\beta$ is strongly-weakly closed. Then,

[TABLE]

and

[TABLE]

Indeed, $\beta_{\nu}^{\prime}(y_{\nu}^{\ast})\rightarrow\beta^{\prime}(y^{\ast})$ a.e., $\int_{0}^{T_{\nu}^{\ast}}\left\|\beta_{\nu}^{\prime}(y_{\nu}^{\ast}(t))\right\|_{L^{2}(\Omega)}dt\leq\int_{0}^{T_{\nu}^{\ast}}\left\|y_{\nu}^{\ast}(t)\right\|_{H_{0}^{1}(\Omega)}^{2}dt\leq C$ and so

[TABLE]

Then, by (6.22)

[TABLE]

and choosing $\varphi\in C_{0}^{\infty}((0,T)\times\Omega)$ with $T>T^{\ast}$ we have

[TABLE]

which yields that $\zeta=\beta^{\prime}(y^{\ast})p$ a.e. Thus, (6.23) holds true. Now, we can pass to the limit in (6.14) and (6.15) to get (6.10) and in (6.13) to deduce

[TABLE]

Finally, we pass to the limit in (6.12), written as $-B^{\ast}p_{\nu}\in N_{\mathcal{K}_{T^{\ast}}}(Pu_{\nu}^{\ast}),$ taking into account that $Pu_{\nu}^{\ast}\rightarrow Pu^{\ast}$ weakly in $L^{2}(0,T^{\ast};H^{-1}(\Omega)),$ $-B^{\ast}p_{\nu}\rightarrow-B^{\ast}p$ strongly in $L^{2}(0,T^{\ast};H^{-1}(\Omega)),$ and $N_{\mathcal{K}_{T^{\ast}}}$ is weakly-strongly closed. Here, $\mathcal{K}_{T^{\ast}}=\{w\in L^{2}(0,T^{\ast};H^{-1}(\Omega));$ $\left\|w(t)\right\|_{H^{-1}(\Omega)}\leq\rho$ a.e. $t\}.\hfill\square$

Example 3. Sliding mode control for reaction-diffusion systems

with nonlinear perturbations.

Let us consider the system

[TABLE]

For certain expressions of $f$ and $g$, equation (6.24) can model different reaction-diffusion processes, as for instance the diffusion, in a habitat $\Omega,$ of two populations with the densities $y$ and $z,$ interacting between them according to the laws expressed by $f$ and $g.$

In some situations, (2.14) can be satisfied and so one can control the first component of the state $y,$ with one controller, letting $z$ uncontrolled. In this example we shall focus on the situation when $V\subset U$ and prove the minimum time sliding mode control for this system.

Case I. Let us consider that $f,$ $g$ are generally nonlinear, $f,$ $g\in C^{2}(\mathbb{R\times R)},$ such that

[TABLE]

and $D_{i}>0,$ $i=1,2.$ We study problem $(\mathcal{P})$ with $U=L^{4}(\Omega).$ We set

[TABLE]

[TABLE]

and $A_{H}:D_{H}\subset H\rightarrow H,$

[TABLE]

In this case,

[TABLE]

with the homogeneous Neumann boundary condition. The controllability $(c_{1})$ can follow as in [8]. Namely, first it is proved that there exists a controller $u\in L^{\infty}((0,\infty)\times\Omega)$ and $T_{\ast},$ such that for $\rho$ large enough $y(T_{\ast})=y_{1}^{\mbox{target}},$ where $T_{\ast}$ depends on $\rho.$ This controller belongs also to $L^{\infty}(0,\infty;L^{4}(\Omega))$ but the controllability follows with a different $\rho$ calculated from a relation between the norms in $L^{4}(\Omega)$ and $L^{\infty}(\Omega).$ Moreover, the time $T_{\ast}$ is smaller as $\rho$ is greater.

Let $K=\{u\in U=L^{4}(\Omega)\times L^{4}(\Omega);$ $\left\|u\right\|_{U}\leq\rho\}.$

**Proposition 6.3. **Assume (6.25) and $y_{0}\in H^{1}(\Omega),$ $y_{1}^{\mbox{target}}\in D,$ $\Delta y_{1}^{\mbox{target}}\in H^{1}(\Omega).$ Then, there exists $T^{\ast},$ $u^{\ast}$ solution to $(\mathcal{P})$ *satisfying *(5.40)-(5.41),

[TABLE]

[TABLE]

where $v=(p,q)$ is the solution to

[TABLE]

Moreover, $y(t)=y_{1}^{\mbox{target}}$ for $t>T^{\ast}.$

**Proof. **It is obvious that $A$ is continuous, monotone and coercive and (2.1)-(2.3), $(a_{2})$ are satisfied. We have for $w:=(y,z)$ and $v:=(p,q)$

[TABLE]

[TABLE]

where $f_{y},$ $f_{z},$ $g_{y},$ $g_{z}$ denote the partial derivatives of $f$ and $g$ with respect to their arguments and they belong to $L^{\infty}(\mathbb{R}\times\mathbb{R}).$ Then,

[TABLE]

and it is easy to see that $(a_{3})-(a_{5})$ are satisfied. Regarding $(a_{6})$ or (2.8) we have for $v=(p,q),$

[TABLE]

where $\mathbf{\Gamma}_{\nu}(p,q)=(\Gamma_{\nu}p,\Gamma_{\nu}q).$

Next, we verify the hypotheses in Section 5.3. Relation (2.12) is satisfied because

[TABLE]

since $H^{1}(\Omega)\subset L^{4}(\Omega)$ for $d\leq 3.$ Also,

[TABLE]

[TABLE]

and

[TABLE]

where $\mathbf{\Gamma}_{H}^{-\alpha}(y,z)=(\Gamma_{H}^{-\alpha}y,\Gamma_{H}^{-\alpha}z).$ Here we used the estimate

[TABLE]

since $L^{2}(\Omega)\subset H^{-\alpha}(\Omega).$ Moreover, $D(\Gamma_{H}^{-\alpha/2})=H^{-\alpha}(\Omega),$ according to the characterization of the domains of the fractionary powers of $-\Delta$ given in [14], Theorem 2, for $\alpha<1.$

Now, we have to check (2.15) and (2.14). To this end, we assume that $\Delta y_{1}^{\mbox{target}}\in H^{1}(\Omega)$ and choose $\widehat{z}$ to be exactly the second component $z$ of the solution to (6.24). First, we prove that it has the necessary regularity. We recall (3) and (3.6), that is

[TABLE]

for all $t\in[0,T],$ $T>0.$ Since $y_{t}\in L^{2}(0,T;H)$ we expect to have a more regular $z.$ We consider the equation

[TABLE]

with $\frac{\partial w}{\partial\nu}=0$ and $w(0)=z_{t}(0)=D_{2}\Delta z_{0}-g(y_{0},z_{0})\in P(V)=H^{1}(\Omega).$ This is computed directly from the equation for $z$, observing that by the hypotheses for $g$ and the initial data we have $g(y_{0}z_{0})\in(I-P)(V)=H^{1}(\Omega).$ We note that (6.26) represents also the equation for $z_{t},$ obtained by formally differentiating the equation in $z.$ Since in (6.26) all coefficients on the left-hand side are in $L^{\infty}(\Omega)$ and $g_{y}(y,z)y_{t}\in L^{2}(0,T;H),$ it follows that it has an unique solution

[TABLE]

and so we can deduce that $w=z_{t}$ and that $z_{t}$ belongs to the same spaces. Moreover, by multiplying (6.26) by $w_{t}$ we obtain, by some calculations similar to those in Theorem 3.2, that

[TABLE]

where this constant depends also on $z_{t}(0)\in P(V),$ namely on $\left\|\Delta z_{0}\right\|_{V},$

[TABLE]

Going back to the equation in $z$ we have

[TABLE]

because $\left\|g(y(t),z(t))\right\|_{H^{1}(\Omega)}\leq C_{T}.$ Indeed, e.g., $\left\|g_{y}(y(t),z(t))\nabla y(t)\right\|_{H^{1}(\Omega)}\leq MC_{T},$ by (3). Next, in the same way we see that if $\Delta y_{1}^{\mbox{target}}\in H^{1}(\Omega),$ then $f(y_{1}^{\mbox{target}},z(t))\in H^{1}(\Omega)$ and so $-D_{1}\Delta y_{1}^{\mbox{target}}+f(y_{1}^{\mbox{target}},z(t))\in H^{1}(\Omega)$. Moreover,

[TABLE]

Here, $C_{0}=\left\|y_{0}\right\|_{H^{1}(\Omega)}+\left\|\Delta z_{0}\right\|_{V}.$ This implies that

[TABLE]

In order to satisfy (2.15) we have to impose that

[TABLE]

We can check that if

[TABLE]

then

[TABLE]

and consequently (6.27) are satisfied. We note that, for any positive constant $C,$ the equation $\sqrt{T}e^{CT}=\frac{1}{C}$ has a unique solution $T_{\ast\ast}$ and so any $T\in[0,T_{\ast\ast})$ verifies (6.28). We can choose $\rho$ sufficiently large, such that the time $T_{\ast}$ in hypothesis $(c_{1})$ becomes smaller enough, such that to remain in $(0,T_{\ast\ast}).$ We have to check (2.14), that is

[TABLE]

which is true for any $t\geq 0,$ in particular for $t\in(0,T_{\ast}+\delta).$

Finally, we prove that $Py(t)=y_{1}^{\mbox{target}}$ for $t\geq T^{\ast}.$ Let us denote the solution to (6.24) for $t\geq T^{\ast}$ by $(\widetilde{y}(t),\widetilde{z}(t)).$ Then, it satisfies

[TABLE]

with homogeneous Neumann boundary conditions and the initial data at $t=T^{\ast},$ $\widetilde{y}(T^{\ast})=y_{1}^{\mbox{target}},$ $\widetilde{z}(T^{\ast})=z(T^{\ast})$. The second equation with $\widetilde{y}=y_{1}^{\mbox{target }}$has a unique solution well defined. If $u$ is replaced in (6.30) by

[TABLE]

where $\widetilde{u}(t)=-D_{1}\Delta y_{1}^{\mbox{target}}+f(y_{1}^{\mbox{target}},\widetilde{z}(t)),$ then $(\widetilde{y}(t),\widetilde{z}(t))=(y_{1}^{\mbox{target}},\widetilde{z}(t))$ verifies the first equation and this proves that the solution slides on the manifold $y_{1}^{\mbox{target}}$ for all $t\geq 0.$ Thus Theorem 5.5 can be applied to obtain the conclusion of the Proposition 6.3. $\hfill\square$

Case II. We can also put into evidence a particular case in which the choice of $\widehat{z}$ is independent of $u,$ $T$ and the system solution. Let us assume that $y_{1}^{\mbox{target}}=0,$ and

[TABLE]

We have to check (2.14). We set $\widehat{y}=(0,\widehat{z}),$ where $\widehat{z}$ in this case can be taken any value such that $A(0,\widehat{z})\in V,$ in particular $\widehat{z}=0.$ We have

[TABLE]

A particular situation is $f(y,z)=yf_{2}(z),$ with $f_{2}(z)$ Lipschitz and positive, for example $f_{2}(z)=\frac{z^{2}}{1+z^{2}}.$

*Case III. Reaction-diffusion systems with linear perturbations. *Let us consider (6.24) with $f(y,z)=a_{1}y+b_{1}z$ and $g(y,z)=a_{2}y+b_{2}z.$ The functional framework is the same as in the precedent example and all hypotheses are satisfied. We shall check only $(d_{4}),$ by setting $\widehat{y}=(y_{1}^{\mbox{target}},z),$ where $z$ is the solution to (6.24) corresponding to $T$ and $u.$ We have

[TABLE]

*Case IV. FitzHugh-Nagumo reaction-diffusion model. *For $f(r_{1},r_{2})=\alpha_{0}r_{1}+r_{2},$ $g(r_{1},r_{2})=-\sigma r_{1}+\gamma r_{2}$ and $D_{2}=0,$ the system (6.24) becomes the well-known FitzHugh-Nagumo model (studied e.g. in [16]). In this case, the hypotheses are verified with the choice

[TABLE]

Example 4. Phase field systems.

Let us consider the phase-field system of Caginalp type, for the phase function $\varphi$ and the energy $\sigma$ written in the following form (see e.g., [8])

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and study $(\mathcal{P})$ with $U=(L^{2}(\Omega),L^{2}(\Omega)),$ $Pu=(u,0),$ $B(u,u_{2})=(u,0)$ and the rest of the spaces as in Example 3. Here $\beta(r)=r^{3}$ and $\pi(r)=-r$ for $r\in\mathbb{R}$, the function $\beta+\pi$ representing the double-well potential. The controllability hypothesis $(c_{1})$ can be proved in a similar way with the proof of [8] for the case $u\in L^{\infty}(0,T;L^{2}(\Omega)),$ $\left\|u(t)\right\|_{2}\leq\rho$ a.e. $t.$ The regularity of the second state component $\varphi$ is proved as in the precedent example and so $\widehat{z}:=\varphi(t)$ which is the appropriate choice for checking (2.15) and (2.14). We mention that the proof of the minimum time for the Caginalp system with the singular logarithmic potential is considered in [10].

Example 5. Diffusion with nonlocal controllers.

We note that the theory works too if** $B$** is a nonlocal operator. This is the case when $B_{i}u_{i}\neq u_{i}.$ Let us consider, for instance Example 1, where $B:L^{2}(\Omega_{1})\rightarrow L^{2}(\Omega)$ is defined by

[TABLE]

and $K\in L^{2}(\Omega\times\Omega_{1}).$ If the kernel $K$ is such that

[TABLE]

it follows that the controllability assumption holds and all conditions are satisfied. Here, $B^{\ast}:L^{2}(\Omega)\rightarrow L^{2}(\Omega_{1})$ is defined by $B^{\ast}v(x)=\int_{\Omega}K(x,z)v(x)dx,$ because

[TABLE]

7 Appendix

Some definitions and results related to operators in Hilbert

spaces.

Let $H,$ $V$ be Hilbert spaces, $V^{\ast}$ the dual of $V$, $V\subset H\subset V^{\ast}$ with compact injections. Let $A:V\rightarrow V^{\ast}$.

The operator $A$ is demicontinuous if $y_{n}\rightarrow y$ strongly in $V$ implies $Ay_{n}\rightarrow Ay$ weakly in $V^{\ast},$ as $n\rightarrow\infty.$

Let $\left\langle\cdot,\cdot\right\rangle_{V^{\ast},V}$ denote the pairing between $V^{\ast}$ and $V.$ The operator $A$ is coercive if

[TABLE]

Let $A$ be an operator on the Hilbert space $H.$ It is called $m$-accretive if it is accretive,

[TABLE]

and $m$-accretive if $R(I+A)=H,$ where $I$ is the identity operator and $R$ is the range. The operator $A$ is quasi $m$-accretive if $(\lambda I+A)$ is $m$-accretive for $\lambda$ sufficiently large.

The operator $A_{H}:D(A_{H})\subset H\rightarrow H$ is the restriction of $A$ on $H$ defined as $A_{H}y=Ay$ for $y\in D(A_{H})=\{y\in V;$ $Ay\in H\}.$

Let $A:V\rightarrow V^{\ast}$ be single-valued, monotone, demicontinuous and coercive. Then, it follows that it is* surjective* (see e.g. [6], p. 36, Corollary 2.2) and $A_{H}$ is $m$-accretive on $H\times H$.

Let $A\in C^{1}(V,V^{\ast}).$ The Gâteaux derivative of $A$ is the linear operator $A^{\prime}(y):V\rightarrow V^{\ast}$ defined by

[TABLE]

If $A_{H}\in C^{1}(D(A_{H}),H)$ we similarly define $(A_{H})^{\prime}(y):D(A_{H})^{\prime}=D(A_{H})\subset H\rightarrow H$ by

[TABLE]

and observe that $(A_{H})^{\prime}(y)=(A^{\prime})_{H}(y),$ for $y\in D(A_{H}),$ so that in the paper we use the notation $A_{H}^{\prime}(y).$

Let $X$ and $Y$ be Banach spaces. The operator $G:X\rightarrow Y$ is said to be strongly continuous from $X$ to $L_{s}(X,Y)$ if for $y_{n}\rightarrow y$ strongly in $X,$ as $n\rightarrow\infty$ it follows

[TABLE]

Duality mapping.

Let $U$ be Banach spaces with the dual $U^{\ast}$ uniformly convex, implying that $U^{\ast}$ and $U$ is reflexive (see e.g., [6], p. 2). Also, it follows that the norm in $U$ is Gâteaux differentiable.

Let $F:U\rightarrow U^{\ast}$ be the duality mapping of $U$, which is single valued and continuous (see e.g., [6], p. 2, Theorem 1.2). We recall that

[TABLE]

Let $K=\{u\in U;$ $\left\|u\right\|_{U}\leq\rho\},$ let $I_{K}$ be the indicator function of $K$ and define

[TABLE]

Then,

[TABLE]

where $\partial j:U\rightarrow U^{\ast}$ is the subdifferential of $j,$ $N_{K}$ is the normal cone to $K$ in $U^{\ast}$ and $\lambda>0.$ The first line in (7.6) should be understood in the multivalued sense.

The conjugate of $j$ is $j^{\ast}:U^{\ast}\rightarrow\mathbb{R}$,

[TABLE]

Then,

[TABLE]

Since $U$ is reflexive, $F^{-1}$ is just the duality mapping of $U^{\ast}$ and so $D(F^{-1})=U^{\ast}$.

If $z\in N_{K}(u),$ then $u\in N_{K}^{-1}(z)$ and we have

[TABLE]

Let $\varepsilon$ be positive and define

[TABLE]

We recall that the subdifferential

[TABLE]

whence

[TABLE]

Then,

[TABLE]

We specify that in the lines before the infimum is realized at $v_{\varepsilon}$ which is the solution to the equation $\varepsilon Fv_{\varepsilon}+N_{K}(v_{\varepsilon})\ni\zeta,$ that is $\varepsilon Fv_{\varepsilon}+z_{\varepsilon}=\zeta,$ where $z_{\varepsilon}\in N_{K}(v_{\varepsilon}).$ Therefore,

[TABLE]

implying by (7.9) that

[TABLE]

Finally, if $u\in(\varepsilon F+N_{K})^{-1}(\zeta),$ it follows that

[TABLE]

and

[TABLE]

The canonical isomorphism

Assume now that $H$ and $V$ are Hilbert spaces, and $V$ has the dual $V^{\ast}.$ The duality mapping, which we denote by $\Gamma:V\rightarrow V^{\ast}$ is the canonical isomorphism of $V$ onto $V^{\ast}$ (see e.g., [6], p. 1). We have

[TABLE]

In addition, $\Gamma_{H},$ the restriction of $\Gamma$ to $H,$ is $m$-accretive on $H\times H,$ with the linear domain denoted $D_{H}$ which is densely, continuously and compactly embedded in $V,$

[TABLE]

For $\nu>0,$ we denote by $\Gamma_{\nu}$ the Yosida approximation of $\Gamma_{H},$ that is

[TABLE]

Comments on the hypothesis of controllability.

Hypothesis $(c_{1})$ ensures that the admissible set for problem $(\mathcal{P})$ is not empty. For example, in the case of Caginalp phase field models the proof of the controllability was provided in [8]. Further, we shall argue for the reliability of such an hypothesis, giving a brief proof of the controllability of (1.1)-(1.2) in some cases. First, let us set

[TABLE]

where Sign$:U^{\ast}\rightarrow 2^{U^{\ast}}$ is defined by

[TABLE]

Here, $B(0,\rho)$ is the ball of center 0 and radius $\rho$ in $U^{\ast}.$ It is well known that $v\rightarrow$Sign $v$ is $m$-accretive on $U^{\ast}$.

Let us consider the problem

[TABLE]

We refer to the case when one state component is controlled by one controller, that is

[TABLE]

and assume

[TABLE]

(When $P=I,$ $Bu=(B_{1}u_{1},B_{2}u_{2}),$ we impose the condition $R(B)=H.$ The proof is the same, by replacing $H_{1}$ by $H.)$

Hypothesis (7.20) implies, by the Banach closed range theorem (see [24], p. 208, Corollary 1) that $(B_{1}^{\ast})^{-1}$ is continuous from $U_{1}$ to $H_{1}$ and $B_{1}^{-1}\in L(H_{1},U_{1}).$ This means $\left\|(B_{1}^{\ast})^{-1}z\right\|_{H_{1}}\leq C\left\|z\right\|_{U_{1}}$ for $z\in U_{1},$ or, equivalently

[TABLE]

We also assume that

[TABLE]

It is clear that when $B_{1}=I,$ then $U_{1}=H_{1}.$ Otherwise, we have the situation in Example 5.

Proposition 7.1. Let $y_{0},$ $Py^{tar}\in P(D_{H})$ and let $(a_{1}),$ $(a_{2}),$ $(b_{1}),$ (7.20),* (7.22) and*

[TABLE]

hold. Then, there exists $T_{\ast}\in(0,T)$ such that, for $\rho$ large enough, $Py(T_{\ast})=Py^{tar},$ where $y$ is the solution to (7.19).

Proof. The operator $B$Sign $(B^{\ast}Pv)$ is $m$-accretive on $H_{1}.$ Indeed, for $v,\bar{v}\in H_{1}$ we have

[TABLE]

because $B^{\ast}Pv=B^{\ast}v,$ $w\rightarrow$Sign $w$ is $m$-accretive and $P^{2}=P$. For the $m$-accretivity let us consider the equation

[TABLE]

which, by denoting $z=y-y^{tar},$ is equivalent with $z+\rho B$Sign $(B^{\ast}z)=f-y^{tar}.$ We set $B^{\ast}z=v\in U_{1}$ and get

[TABLE]

Denoting $G=B^{-1}(B^{\ast})^{-1}$ we see that $G\in L(U_{1},U_{1})$ and $(Gv,v)_{U_{1}}=\left\|(B^{\ast})^{-1}v\right\|_{H_{1}}^{2}\geq\left\|v\right\|_{U_{1}}^{2}$ (because $B^{\ast}$ is continuous, see (7.21)$).$ Now, Sign $w$ is $m$-accretive on $U_{1}\times U_{1}$, hence $R(G+$Sign$)=U_{1}$ (see e.g., [6], p. 44, Corollary 2.6).

Then, we prove that (7.19) has a unique solution.

Since $A_{H}$ is quasi $m$-accretive and $S=\rho B$Sign $(B^{\ast}P(y(t)-y^{tar}))$ is $m$-accretive with $D(S)=H_{1}$ and $D_{H}\cap\overset{\circ}{D(S)}=D_{H}\neq\varnothing,$ it follows by that $A_{H}+S$ is quasi $m$-accretive on $H\times H$ (see [6], p. 43, Theorem 2.6). Therefore, (7.19) has a unique solution $y\in L^{\infty}(0,T;D_{H})\cap W^{1,\infty}(0,T;H)$ satisfying estimate (3) (see the proof of Theorem 3.2).

Now, we can justify the controllability assertion. Let us write (7.19) in the equivalent form

[TABLE]

and multiply it by $P(y(t)-y^{tar}).$ By (7.22) we get

[TABLE]

Further, we obtain

[TABLE]

Next, we use (7.21) for $w=P(y(t)-y^{tar})$ which implies $\left\|B^{\ast}P(y(t)-y^{tar})\right\|_{U_{1}}\geq C\left\|P(y(t)-y^{tar})\right\|_{H_{1}}$ and so

[TABLE]

For $\rho>\left\|A_{H}y^{tar}\right\|_{H},$ this yields

[TABLE]

Finally, we obtain that $t\rightarrow\left\|P(y(t)-y^{tar})\right\|_{H}$ is strictly decreasing, vanishes at $t=T_{\ast}$ below and the previous relation takes place for

[TABLE]

for $\rho>\left\|A_{H}y^{tar}\right\|_{H}+C_{1}\left\|P(y_{0}-y^{tar})\right\|_{H}.$ We also observe that $T^{\ast}$ decreases as $\rho$ increases and in fact $T_{\ast}\rightarrow 0$ as $\rho\rightarrow\infty.$ This end the proof. $\square$

**Acknowledgement. **The present paper benefits from the support of the of Ministry of Research and Innovation, CNCS –UEFISCDI, project number PN-III-P4-ID-PCE-2016-0011.