Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems

TL;DR

This paper develops a dynamic programming framework for fractional-order control systems, establishing a Hamilton-Jacobi-Bellman equation and methods for constructing optimal feedback controls.

Contribution

It introduces a new notion of coinvariant derivatives for fractional systems and links the value functional to a HJB equation under smoothness assumptions.

Findings

Proves the dynamic programming principle for fractional systems

Establishes a connection between the value functional and HJB equation

Proposes a method for constructing optimal feedback controls

Abstract

We consider a Bolza-type optimal control problem for a dynamical system described by a fractional differential equation with the Caputo derivative of an order $\alpha \in (0, 1)$. The value of this problem is introduced as a functional in a suitable space of histories of motions. We prove that this functional satisfies the dynamic programming principle. Based on a new notion of coinvariant derivatives of the order $\alpha$, we associate the considered optimal control problem with a Hamilton-Jacobi-Bellman equation. Under certain smoothness assumptions, we establish a connection between the value functional and a solution to this equation. Moreover, we propose a way of constructing optimal feedback controls. The paper concludes with an example.

Equations

$$(I^{\alpha }\psi )(\tau )=\frac{1}{\Gamma (\alpha )}{\int }_0^{\tau }\frac{\psi (\xi )}{(\tau -\xi {)}^{1-\alpha }}\mathrm{d}\xi ,\tau \in [0,t],$$

$$\Vert (I^{\alpha }\psi )(\tau )-(I^{\alpha }\psi )({\tau }^{\prime })\Vert \le H_{\alpha }\Vert \psi (\cdot ){\Vert }_{[0,t]}|\tau -{\tau }^{\prime }{|}^{\alpha },\tau ,{\tau }^{\prime }\in [0,t].$$

$$x(\tau )=x(0)+(I^{\alpha }\psi )(\tau ),\tau \in [0,t],$$

$$\big{(}I^{1-\alpha}(x(\cdot)-x(0))\big{)}(\tau)=\big{(}I^{1-\alpha}(I^{\alpha}\psi)\big{)}(\tau)=\int_{0}^{\tau}\psi(\xi)\,\mathrm{d}\xi,\quad\tau\in[0,t].$$

$$(^{C}D^{\alpha}x)(\tau)=\frac{\mathrm{d}}{\mathrm{d}\tau}\big{(}I^{1-\alpha}(x(\cdot)-x(0))\big{)}(\tau)=\frac{1}{\Gamma(1-\alpha)}\frac{\mathrm{d}}{\mathrm{d}\tau}\int_{0}^{\tau}\frac{x(\xi)-x(0)}{(\tau-\xi)^{\alpha}}\,\mathrm{d}\xi$$

$$\displaystyle x(\tau)=x(0)+\big{(}I^{\alpha}(^{C}D^{\alpha}x)\big{)}(\tau),$$

$$\displaystyle\big{(}I^{1-\alpha}(x(\cdot)-x(0))\big{)}(\tau)=\int_{0}^{\tau}(^{C}D^{\alpha}x)(\xi)\,\mathrm{d}\xi$$

$${(}^C{D}^{\alpha }x)(\tau )=f(\tau ,x(\tau ),u(\tau )),x(\tau )\in {\mathbb{R}}^n,u(\tau )\in P,\tau \in [0,T],$$

$$x(0)=w_0.$$

$$\Vert f(\tau ,x,u)-f(\tau ,x^{\prime },u)\Vert \le {\lambda }_f\Vert x-x^{\prime }\Vert ,\tau \in [0,T],x,x^{\prime }\in B(R),u\in P,$$

$$\Vert f(\tau ,x,u)\Vert \le (1+\Vert x\Vert )c_f,\tau \in [0,T],x\in {\mathbb{R}}^n,u\in P.$$

$$x(\tau )=w_0+\frac{1}{\Gamma (\alpha )}{\int }_0^{\tau }\frac{f(\xi ,x(\xi ),u(\xi ))}{(\tau -\xi {)}^{1-\alpha }}\mathrm{d}\xi ,\tau \in [0,T].$$

$$J(0,w_0,u(\cdot ))=\sigma (x(T))+{\int }_0^T\chi (\tau ,x(\tau ),u(\tau ))\mathrm{d}\tau ,$$

$$\rho (0,w_0)=\underset{u(\cdot )\in \mathcal{U}}{\inf }J(0,w_0,u(\cdot )).$$

$$J(0,w_0,u^{(\epsilon )}(\cdot ))\le \rho (0,w_0)+\epsilon .$$

$${(}^C{D}^{\alpha }x)(\tau )=\Gamma (\alpha +1)u(\tau ),x(\tau )\in \mathbb{R},|u(\tau )|\le 1,\tau \in [0,2],$$

$$x(0)=w_0=2^{\alpha -1}+1$$

$$J(0,w_0,u(\cdot ))=x^2(2).$$

$${(}^C{D}_1^{\alpha }y)(\tau )=\Gamma (\alpha +1)u(\tau ),y(\tau )\in \mathbb{R},|u(\tau )|\le 1,\tau \in [1,2],$$

$$y(1)=x^{\circ }(1)=2^{\alpha -1}$$

$$J(1,x^{\circ }(1),u(\cdot ))=y^2(2).$$

$${(}^C{D}_1^{\alpha }y)(\tau )=\frac{1}{\Gamma (1-\alpha )}\frac{\mathrm{d}}{\mathrm{d}\tau }{\int }_1^{\tau }\frac{y(\xi )-y(1)}{(\tau -\xi {)}^{\alpha }}\mathrm{d}\xi ,\tau \in [1,2].$$

$$y^{\ast }(\tau )=x^{\circ }(1)+\frac{1}{\Gamma (\alpha )}{\int }_1^{\tau }\frac{\Gamma (\alpha +1)u^{\ast }(\xi )}{(\tau -\xi {)}^{1-\alpha }}\mathrm{d}\xi =2^{\alpha -1}-(\tau -1{)}^{\alpha },\tau \in [1,2],$$

$$\bar{u}(\tau )=\{\begin{array}{ll}-1,& \text{mbox}if\tau \in [1,\theta ),\\ 0,& \text{mbox}if\tau \in [\theta ,2],\end{array}\theta =2-(1-2^{\alpha -1}{)}^{1/\alpha },$$

$$x(\tau )=w^{\circ }(\tau ),\tau \in [0,1].$$

$$J(1,w^{\circ }(\cdot ),u(\cdot ))=x^2(2).$$

$$x(\tau )=w^{\circ }(0)+\frac{1}{\Gamma (\alpha )}{\int }_0^1\frac{{(}^C{D}^{\alpha }{w}^{\circ })(\xi )}{(\tau -\xi {)}^{1-\alpha }}\mathrm{d}\xi +\frac{1}{\Gamma (\alpha )}{\int }_1^{\tau }\frac{\Gamma (\alpha +1)u(\xi )}{(\tau -\xi {)}^{1-\alpha }}\mathrm{d}\xi .$$

$$x(\tau )=\tilde{w}(\tau ),\tau \in [0,1],$$

$$J(1,\tilde{w}(\cdot ),u(\cdot ))=x^2(2),$$

$$x(\tau )=w(\tau ),\tau \in [0,t],$$

$$x(\tau )=w(0)+\frac{1}{\Gamma (\alpha )}{\int }_0^t\frac{{(}^C{D}^{\alpha }w)(\xi )}{(\tau -\xi {)}^{1-\alpha }}\mathrm{d}\xi +\frac{1}{\Gamma (\alpha )}{\int }_t^{\tau }\frac{f(\xi ,x(\xi ),u(\xi ))}{(\tau -\xi {)}^{1-\alpha }}\mathrm{d}\xi .$$

Full text

{AMS}

26A33, 34A08, 49L20, 35F21

1 Introduction

The dynamic programming principle, along with the Pontryagin maximum principle, is one of the fundamental and most effective tools for studying and constructing solutions of various optimal control problems. The goal of the paper is to extend the dynamic programming principle to the case of fractional-order dynamical systems. More precisely, we focus on a Bolza-type optimal control problem for a system described by a fractional differential equation with the Caputo derivative of an order $\alpha\in(0,1)$. Such problems arise in a wide range of applications, including, e.g., biology [42], chemistry [12], economics [17], electrical engineering [19], and medicine [22]. For the basics of fractional calculus and the theory of fractional-order differential equations, the reader is referred to, e.g., [38, 23, 10].

In order to formulate the dynamic programming principle, for every intermediate time $t\in(0,T)$, it is necessary to introduce an auxiliary optimal control problem (sub-problem) with this time $t$ considered as the initial one. This auxiliary problem should be consistent with the original one in the sense that the optimality principle (see, e.g., [6, § 3]) should be satisfied. The analysis of simple examples shows (see Section 4 for a detailed discussion) that this sub-problem should be formulated for the original dynamical system but with the whole history of the motion $x_{t}(\tau)=x(\tau)$, $\tau\in[0,t]$, taken as the initial data at the time $t$, and, hence, the pair $(t,x_{t}(\cdot))$ should be regarded as a position of the system formed by the time $t$. This is explained by the fact that the Caputo derivative has a nonlocal character (see, e.g., [41]), i.e., its value $(^{C}D^{\alpha}x)(t)$ at the time $t$ depends essentially on all previous values $x(\tau)$, $\tau\in[0,t]$. Thus, we conclude that, in the optimal control problem under consideration, the value (the optimal result) should be introduced as a functional $\rho(t,w(\cdot))$, where the function $w(\tau),$ $\tau\in[0,t]$, is treated as a history of a motion of the system on $[0,t]$. This circumstance significantly differs the approach developed in the paper from the existing works on the dynamic programming principle for fractional-order systems (see, e.g., [18, 36, 37]), in which the value is a function $\rho(t,x)$, where $x$ is treated as a value of the state vector at the time $t$. Let us note that this approach, involving the dependence of the value on a history of a motion, is generally accepted in the control theory for functional-differential systems, and, therefore, due to the relationship between fractional-order systems and functional-differential systems of a neutral type (see, e.g., [15, Sect. 4]), it seems natural to use this approach in the considered problem, too.

In many control problems, for finding the value and constructing optimal feedback controls, it is often more convenient to apply not the dynamic programming principle itself, but its infinitesimal form expressed in terms of a Hamilton–Jacobi–Bellman equation. The second goal of the paper is to derive and study a Hamilton–Jacobi–Bellman equation that corresponds to the considered optimal control problem.

In order to do this, we introduce a new notion of coinvariant ($ci$-) differentiation of the order $\alpha$ of functionals $\varphi(t,w(\cdot))$, where $(t,w(\cdot))$ is a position of the system from a suitable functional space. This notion naturally agrees (when, formally, $\alpha=1$) with the notion of $ci$-differentiation, which has proved to be a convenient tool for developing the theory of Hamilton–Jacobi equations for various control problems in functional-differential systems of retarded (see, e.g., [25, 29, 30, 31, 20, 35] and the references therein) and neutral (see, e.g., [32, 16]) types (see also [21]). Let us note that, e.g., in [4, 11, 34, 5, 39] (see also the references therein), some other notions of differentiation of functionals are used, which, in some sense, are close to $ci$-differentiation. The main advantage of the introduced notion of fractional $ci$-differentiation is the fact that it allows us to obtain a simple formula for the total derivative of the value functional along a motion of the fractional-order system.

Based on this formula and the dynamic programming principle, we associate the considered optimal control problem with the Hamilton–Jacobi–Bellman equation with the $ci$-derivatives of the order $\alpha$ and establish a connection between them. Namely, we show that, if the value functional is sufficiently smooth ($ci$-smooth of the order $\alpha$), then it satisfies this equation. On the other hand, we prove that if the Cauchy problem for this equation and a natural right-end condition admits a $ci$-smooth of the order $\alpha$ solution, then it coincides with the value functional, and, moreover, we can construct an optimal feedback control strategy by using the extremal shift in the direction of the $ci$-gradients of the order $\alpha$ of this solution. Here, we use a quite general notion of feedback control strategies that goes back to the positional approach in differential games [28, 26] (see also [33, 29, 30, 31] and [13, 15]). Finally, we illustrate the obtained results by an example.

The rest of the paper is organized as follows. In Section 2, we introduce the notations, recall the definitions of fractional-order integrals and derivatives, and give some of their properties. In Section 3, we describe the optimal control problem under consideration. Section 4 is devoted to a discussion of the dynamic programming principle for fractional-order systems on a simple example. Further, in Section 5, a general statement of the optimal control problem with an arbitrary initial position $(t,w(\cdot))$ is given, and the value functional is defined. The dynamic programming principle is proved in Section 6. In Sections 7 and 8, we establish some technical properties of motions of the system and show that the value functional is continuous. In Section 9, we introduce the notion of $ci$-differentiability of the order $\alpha$. Sections 10 and 11 deal with the Hamilton–Jacobi–Bellman equation and its connection with the considered optimal control problem. Based on the obtained results, in Section 12, we solve the example from Section 4. Concluding remarks are given in Section 13.

2 Notations and definitions

Let $T>0$, $n\in\mathbb{N}$, and $\alpha\in(0,1)$ be fixed throughout the paper. By $\|\cdot\|$ and $\langle\cdot,\cdot\rangle$, we denote the Euclidian norm and the inner product in $\mathbb{R}^{n}$.

For every $t\in[0,T]$, let $L^{\infty}([0,t],\mathbb{R}^{n})$ be the set of (Lebesgue) measurable functions $\psi:[0,t]\rightarrow\mathbb{R}^{n}$ such that $\|\psi(\cdot)\|_{[0,t]}<\infty$, where $\|\psi(\cdot)\|_{[0,t]}=\operatorname{ess}\sup_{\tau\in[0,t]}\|\psi(\tau)\|$ if $t>0$ and $\|\psi(\cdot)\|_{[0,t]}=\|\psi(0)\|$ if $t=0$. For a function $\psi(\cdot)\in L^{\infty}([0,t],\mathbb{R}^{n})$, the (left-sided) Riemann–Liouville fractional integral of the order $\alpha$ is defined by

[TABLE]

where $\Gamma$ is the gamma function. According to, e.g., [38, Theorem 3.6 and Remark 3.3] (see also [10, Theorem 2.6]), for $H_{\alpha}=2/\Gamma(\alpha+1)$, we have

[TABLE]

Let us denote by ${AC^{\alpha}}([0,t],\mathbb{R}^{n})$ the set of functions $x:[0,t]\rightarrow\mathbb{R}^{n}$ that can be represented in the form

[TABLE]

for some function $\psi(\cdot)\in L^{\infty}([0,t],\mathbb{R}^{n})$. This set ${AC^{\alpha}}([0,t],\mathbb{R}^{n})$ is considered as a subset of the space ${C}([0,t],\mathbb{R}^{n})$ of continuous functions endowed with the uniform norm $\|\cdot\|_{[0,t]}$. In the case $t=0$, the set ${AC^{\alpha}}([0,t],\mathbb{R}^{n})$ can be identified with $\mathbb{R}^{n}$.

Let $x(\cdot)\in{AC^{\alpha}}([0,t],\mathbb{R}^{n}),$ and let $\psi(\cdot)\in L^{\infty}([0,t],\mathbb{R}^{n})$ be such that Eq. 2 is valid. Then, according to, e.g., [38, Theorem 2.5] (see also [10, Theorem 2.2]), we obtain

[TABLE]

Hence, the (left-sided) Caputo fractional derivative of $x(\cdot)$ of the order $\alpha$ defined by

[TABLE]

exists for almost every (a.e.) $\tau\in[0,t]$, and, moreover, $(^{C}D^{\alpha}x)(\tau)=\psi(\tau)$ for a.e. $\tau\in[0,t]$. In particular, we have $\|(^{C}D^{\alpha}x)(\cdot)\|_{[0,t]}=\|\psi(\cdot)\|_{[0,t]}<\infty$, and, from Eqs. 2 and 3, it follows that the equalities

[TABLE]

hold for every $\tau\in[0,t]$.

3 Optimal control problem

Let us consider a dynamical system which motion is described by the fractional differential equation

[TABLE]

with the initial condition

[TABLE]

Here, $\tau$ is time; $x(\tau)$ and $u(\tau)$ are the current values of the state and control vectors, respectively; $(^{C}D^{\alpha}x)(\tau)$ is the Caputo derivative of the order $\alpha$ (see Eq. 4); $P\subset\mathbb{R}^{n_{u}}$ is a compact set, $n_{u}\in\mathbb{N}$; $w_{0}\in\mathbb{R}^{n}$ determines the initial value of the state vector.

{assumption}

The function $f:[0,T]\times\mathbb{R}^{n}\times P\rightarrow\mathbb{R}^{n}$ in Eq. 7 satisfies the following conditions: (a) $f$ is continuous; (b) $f$ is locally Lipschitz continuous in the second argument, i.e., for any $R>0,$ there exists $\lambda_{f}>0$ such that

[TABLE]

where we denote $B(R)=\{x\in\mathbb{R}^{n}:\,\|x\|\leq R\}$; (c) $f$ has sublinear growth in the second argument, i.e., there exists $c_{f}>0$ such that

[TABLE]

By an admissible control $u(\cdot)$, we mean a measurable function $u:[0,T]\rightarrow P$. The set of all such controls is denoted by $\mathcal{U}$. A motion of system Eq. 7 with initial condition Eq. 8 that corresponds to a control $u(\cdot)\in\mathcal{U}$ is a function $x(\cdot)\in{AC^{\alpha}}([0,T],\mathbb{R}^{n})$ that satisfies the equality in Eq. 8 and, together with $u(\cdot)$, the differential equation in Eq. 7 for a.e. $\tau\in[0,T]$. Due to Section 3, such a motion $x(\cdot)$ exists and is unique (see, e.g., [14, Proposition 5.1]). Moreover, $x(\cdot)$ is a unique function from ${C}([0,T],\mathbb{R}^{n})$ that satisfies the integral equation

[TABLE]

The goal of control is to minimize the cost functional

[TABLE]

where $x(\cdot)$ is the motion of system Eq. 7 with initial condition Eq. 8 that corresponds to a control $u(\cdot)\in\mathcal{U}$. The first argument in the notation $J(0,w_{0},u(\cdot))$ is added to indicate the initial time, which is convenient for further constructions (see Section 5).

{assumption}

The functions $\sigma:\mathbb{R}^{n}\rightarrow\mathbb{R}$ and $\chi:[0,T]\times\mathbb{R}^{n}\times P\rightarrow\mathbb{R}$ in Eq. 11 are continuous.

The value of the optimal control problem Eqs. 7, 8, and 11 is defined by

[TABLE]

A control $u^{\circ}(\cdot)\in\mathcal{U}$ is called optimal if $J(0,w_{0},u^{\circ}(\cdot))=\rho(0,w_{0})$. However, since the problem is considered under rather general assumptions, one can not expect that an optimal control exists. Therefore, we are interested in finding, for every sufficiently small $\varepsilon>0$, an $\varepsilon$-optimal control $u^{(\varepsilon)}(\cdot)\in\mathcal{U}$, which satisfies the inequality

[TABLE]

The goal of the paper is to extend the dynamic programming principle to the optimal control problem Eqs. 7, 8, and 11 and, furthermore, derive the corresponding Hamilton–Jacobi–Bellman equation.

The dynamic programming principle is based on the principle of optimality (see, e.g., [6, § 3]), which, in particular, states the following. An optimal control $u^{\circ}(\cdot)$ has the property that, for any intermediate time $t\in(0,T)$, the control $u^{\ast}(\tau)=u^{\circ}(\tau)$, $\tau\in[t,T]$, must be optimal for the sub-problem on the remaining time interval $[t,T]$ with the initial position resulting from the previous control $u_{\ast}(\tau)=u^{\circ}(\tau)$, $\tau\in[0,t]$. Thus, in order to apply this principle to the problem Eqs. 7, 8, and 11, it is necessary to define what is meant by “sub-problem” and by “position”. One can propose several quite reasonable ways to answer this question. In the next section, we consider a simple example to illustrate three of them. Let us note that a complete solution to this example is given in Section 12 on the basis of the results of the paper.

4 Example

Let the optimal control problem be described by the system

[TABLE]

with the initial condition

[TABLE]

and the cost functional

[TABLE]

In Eq. 12, the coefficient $\Gamma(\alpha+1)$ is added to simplify the formulas below.

By direct calculations, one can show that $u^{\circ}(\tau)=-1$, $\tau\in[0,2]$, is the optimal control, and the corresponding motion of Eqs. 12 and 13 is $x^{\circ}(\tau)=2^{\alpha-1}+1-\tau^{\alpha}>0$, $\tau\in[0,2]$. In particular, we have $\rho(0,w_{0})=(1-2^{\alpha-1})^{2}>0$.

Taking into account the discussion of the optimality principle in Section 3, let us choose $t=1$ as an intermediate time and propose the following three approaches to defining the appropriate notions of “sub-problem” and “position”.

4.1 First approach

As a sub-problem on the time interval $[1,2]$, let us consider the optimal control problem for the system

[TABLE]

with the initial condition

[TABLE]

and the cost functional

[TABLE]

Here, by $(^{C}D_{1}^{\alpha}y)(\tau)$, we denote the Caputo derivative (see Eq. 4) with the lower terminal $1$, which is defined by

[TABLE]

Thus, this sub-problem is the same as the original one Eqs. 12, 13, and 14 but with the initial time formally changed to $1$, and the role of the position resulting from the control $u_{\ast}(\tau)=u^{\circ}(\tau)=-1$, $\tau\in[0,1]$, is played by the pair $(1,x^{\circ}(1))$.

Let us show that the optimality principle is not satisfied within this approach, i.e., the control $u^{\ast}(\tau)=u^{\circ}(\tau)=-1$, $\tau\in[1,2]$, is not optimal in the problem Eqs. 15, 16, and 17. For the motion $y^{\ast}(\cdot)$ of Eqs. 15 and 16 that corresponds to $u^{\ast}(\cdot)$, we have

[TABLE]

and, therefore, $J(1,x^{\circ}(1),u^{\ast}(\cdot))=(2^{\alpha-1}-1)^{2}>0$. However, choosing the control

[TABLE]

for the corresponding motion $\bar{y}(\cdot)$ of Eqs. 15 and 16, we obtain $\bar{y}(2)=0$, and, hence, $J(1,x^{\circ}(1),\bar{u}(\cdot))=0<J(1,x^{\circ}(1),u^{\ast}(\cdot))$. Thus, the control $u^{\ast}(\cdot)$ is not optimal.

4.2 Second approach

It follows from the previous arguments that we can not simply change the original system by substituting the fractional derivative with another lower terminal. Taking this into account, in a sub-problem, let us consider the same system Eq. 12. According to Eq. 4, the value of the fractional derivative $(^{C}D^{\alpha}x)(\tau)$ for $\tau\in[0,2]$ depends on the values $x(\xi)$ for all $\xi\in[0,\tau]$. Hence, at least formally, in order to correctly formulate an initial value problem for the differential equation in Eq. 12 with the initial time $t=1$, one should specify the values $x(\xi)$ for all $\xi\in[0,1]$. However, all these values are known since they have already been realized during the time interval $[0,1]$. Then, if we denote the history of the motion $x^{\circ}(\cdot)$ on $[0,1]$ by $w^{\circ}(\tau)=x^{\circ}(\tau)$, $\tau\in[0,1]$, we obtain the following initial condition:

[TABLE]

Thus, we come to the optimal control problem (sub-problem) for system Eq. 12 with initial condition Eq. 20 and the cost functional

[TABLE]

Here, in accordance with Eq. 10, the motion $x(\cdot)$ for $\tau\in[1,2]$ is given by

[TABLE]

Nevertheless, one might assume that a solution to the described sub-problem depends not on the whole history $w^{\circ}(\tau)$, $\tau\in[0,1]$, of the motion $x^{\circ}(\cdot)$, but only on the single value $w^{\circ}(1)=x^{\circ}(1)$, and, therefore, the pair $(1,x^{\circ}(1))$ still plays the role of the position (as in Section 4.1). In general, this assumption may be motivated by a specific form of system Eq. 12 and/or more subtle results on the uniqueness of solutions of fractional differential equations (see, e.g., [9]). However, let us show that the optimality principle is not satisfied within this approach too, i.e., there exists another history $\tilde{w}(\cdot)$ such that $\tilde{w}(1)=x^{\circ}(1)$ but the control $u^{\ast}(\tau)=u^{\circ}(\tau)=-1$, $\tau\in[1,2]$, is not optimal in the problem for system Eq. 12 with the initial condition

[TABLE]

and the cost functional

[TABLE]

where $x(\cdot)$ is defined by analogy with Eq. 22. Namely, let $\tilde{w}(\tau)=x^{\circ}(1)=2^{\alpha-1}$, $\tau\in[0,1]$. Let us note that this function can be considered as the history on $[0,1]$ of the motion $\tilde{x}(\cdot)$ of system Eq. 12 that corresponds to the initial condition $\tilde{x}(0)=2^{\alpha-1}$ and the control $\tilde{u}(\tau)=0$, $\tau\in[0,2]$. As in Eq. 18, for the motion $x^{\ast}(\cdot)$ of Eqs. 12 and 23 that corresponds to $u^{\ast}(\cdot)$, we have $x^{\ast}(\tau)=2^{\alpha-1}-(\tau-1)^{\alpha}$, $\tau\in[1,2]$, and, hence, $J(1,\tilde{w}(\cdot),u^{\ast}(\cdot))=(2^{\alpha-1}-1)^{2}>0$. On the other hand, for the motion $\bar{x}(\cdot)$ of Eqs. 12 and 23 that corresponds to $\bar{u}(\cdot)$ from Eq. 19, we obtain $\bar{x}(2)=0$, and, therefore, $J(1,\tilde{w}(\cdot),\bar{u}(\cdot))=0<J(1,\tilde{w}(\cdot),u^{\ast}(\cdot))$.

4.3 Third approach

Summarizing the above, we conclude that, as the position, we should consider the pair $(t,w^{\circ}(\cdot))$ with the whole history $w^{\circ}(\cdot)$ of the motion $x^{\circ}(\cdot)$ on the time interval $[0,t]$. In fact, it follows from Theorem 6.1 below that the optimality principle is satisfied with the sub-problem defined by Eqs. 12, 20, and 21. Let us note that this approach goes back to the control theory for functional-differential systems. For more details on the relationship between fractional-order systems and functional-differential systems of a neutral type, the reader is referred to, e.g., [15, Sect. 4].

5 Optimal control problem: statement for an arbitrary position

Thus, in accordance with Section 4 (see also [13, 15]), by a position of system Eq. 7, we mean a pair $(t,w(\cdot))$ consisting of a time $t\in[0,T]$ and a function $w(\cdot)\in{AC^{\alpha}}([0,t],\mathbb{R}^{n})$ (see the definition of ${AC^{\alpha}}([0,t],\mathbb{R}^{n})$ in Section 2), which is treated as a history of a motion $x(\cdot)$ of system Eq. 7 on the time interval $[0,t]$. The set of all such positions $(t,w(\cdot))$ is denoted by $G$.

In this section, for an arbitrary position $(t,w(\cdot))\in G$ considered as the initial one, we formulate the optimal control problem (sub-problem) that is consistent with the original one Eqs. 7, 8, and 11. Besides, we introduce some auxiliary notations.

Let $(t,w(\cdot))\in G$ be fixed, and let $\theta\in[t,T]$. The set $\mathcal{U}(t,\theta)$ of admissible controls on $[t,\theta]$ consists of all measurable functions $u:[t,\theta]\rightarrow P$. Due to Section 3, for the position $(t,w(\cdot))$ and a control $u(\cdot)\in\mathcal{U}(t,\theta)$, there exists a unique motion of system Eq. 7, which is a function $x(\cdot)\in{AC^{\alpha}}([0,\theta],\mathbb{R}^{n})$ that satisfies the equality

[TABLE]

and, together with $u(\cdot)$, the differential equation in Eq. 7 for a.e. $\tau\in[t,\theta]$ (see, e.g., [15, Proposition 2]). This motion $x(\cdot)$ is a unique function from ${C}([0,\theta],\mathbb{R}^{n})$ that satisfies Eq. 24 and, for every $\tau\in[t,\theta]$, the integral equation

[TABLE]

Below, for the motion $x(\cdot)$, we also use the notation $x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$. Besides, for every $\tau\in[0,\theta]$, by $x_{\tau}(\cdot)$, we denote the history of the motion $x(\cdot)$ on $[0,\tau]$ defined by

[TABLE]

Let us note that the inclusions $(\tau,x_{\tau}(\cdot))\in G$, $\tau\in[0,\theta]$, are valid. Moreover, let us emphasize that, under the considered statement, motions of system Eq. 7 satisfy the so-called semigroup property (see, e.g., [15, Sect. 3.2]).

In the optimal control problem for system Eq. 7 with the initial position $(t,w(\cdot))$, by choosing a control $u(\cdot)\in\mathcal{U}(t,T)$, we want to minimize the cost functional

[TABLE]

where $x(\cdot)=x(\cdot\mid t,w(\cdot),T,u(\cdot))$. Thus, the value of this problem is given by

[TABLE]

and we are interested in finding, for every $\varepsilon>0$, an $\varepsilon$-optimal control $u^{(\varepsilon)}(\cdot)\in\mathcal{U}(t,T)$:

[TABLE]

Let us note that, in the case $t=0$ and $w(0)=w_{0}$, the statement of the considered optimal control problem for an arbitrary initial position $(t,w(\cdot))\in G$ agrees with the original one given in Section 3.

6 Dynamic programming principle

Since relation Eq. 28 defines the value $\rho(t,w(\cdot))$ for every $(t,w(\cdot))\in G$, we can consider the value functional $\rho:G\rightarrow\mathbb{R}$. We use the term “functional” to emphasize that $\rho(t,w(\cdot))$ depends on the infinite-dimensional argument $w(\cdot)$, which is consistent with the discussion in Section 4.

The value functional $\rho$ satisfies the terminal condition

[TABLE]

and the following dynamic programming principle.

Theorem 6.1**.**

For any $(t,w(\cdot))\in G$ and $\theta\in[t,T]$, the equality below holds:

[TABLE]

*where $x(\cdot)=x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$ is the motion of system Eq. 7. *

Proof 6.2**.**

The proof follows the standard scheme (see, e.g., [44, Theorem 2.4.2]). Let $(t,w(\cdot))\in G$ and $\theta\in[t,T]$ be fixed. Let us denote the right-hand side of the equality in (31) by $\bar{\rho}$ and prove that $\rho(t,w(\cdot))=\bar{\rho}$.

For every $u(\cdot)\in\mathcal{U}(t,\theta)$ and $u^{\ast}(\cdot)\in\mathcal{U}(\theta,T)$, let us consider $\bar{u}(\cdot)\in\mathcal{U}(t,T)$ such that $\bar{u}(\tau)=u(\tau)$, $\tau\in[t,\theta)$, and $\bar{u}(\tau)=u^{\ast}(\tau)$, $\tau\in[\theta,T]$. Then, we have

[TABLE]

where $x(\cdot)=x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$. Taking the infimum over $u^{\ast}(\cdot)\in\mathcal{U}(\theta,T)$, we obtain

[TABLE]

Further, taking the infimum over $u(\cdot)\in\mathcal{U}(t,\theta)$, we get $\rho(t,w(\cdot))\leq\bar{\rho}$.

On the other hand, let $\varepsilon>0$, and let $u^{(\varepsilon)}(\cdot)\in\mathcal{U}(t,T)$ be an $\varepsilon$-optimal control (see Eq. 29). Let us define $u(\tau)=u^{(\varepsilon)}(\tau)$, $\tau\in[t,\theta]$, and $u^{\ast}(\tau)=u^{(\varepsilon)}(\tau)$, $\tau\in[\theta,T]$, and consider the motion $x(\cdot)=x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$ of system Eq. 7. Hence, we derive

[TABLE]

*Since this estimate is valid for every $\varepsilon>0$, then $\rho(t,w(\cdot))\geq\bar{\rho}$. Thus, we have $\rho(t,w(\cdot))=\bar{\rho}$, and the theorem is proved. *

The dynamic programming principle gives a nonlocal characterization of the value functional $\rho$. However, in order to find this functional and, based on this, construct $\varepsilon$-optimal controls, it is often more convenient to apply an infinitesimal form of this principle, which is expressed in terms of a Hamilton–Jacobi–Bellman equation. Before proceeding to the derivation and study of the Hamilton–Jacobi–Bellman equation associated with the optimal control problem Eqs. 7 and 11, we establish some technical properties of motions of system Eq. 7 and prove that the value functional $\rho$ is continuous.

7 Properties of motions of the system

The goal of this section is to prove the proposition below, which states the uniform boundedness and Hölder continuity of motions of system Eq. 7, and also their Lipschitz dependence on the initial positions.

Proposition 7.1**.**

For any $R>0$, the following statements hold:

• $i)$

there exists $M_{x}>0$ such that, for any $(t,w(\cdot))\in G$ satisfying $\|w(\cdot)\|_{[0,t]}\leq R$, any $\theta\in[t,T]$, and any $u(\cdot)\in\mathcal{U}(t,\theta)$, the inequality $\|x(\cdot)\|_{[0,\theta]}\leq M_{x}$ is valid for the motion $x(\cdot)=x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$ of system Eq. 7;

• $ii)$

there exists $H_{x}>0$ such that, for any $(t,w(\cdot))\in G$ satisfying $\|w(\cdot)\|_{[0,t]}\leq R$ and $\|(^{C}D^{\alpha}w)(\cdot)\|_{[0,t]}\leq R$, any $\theta\in[t,T]$, and any $u(\cdot)\in\mathcal{U}(t,\theta)$, for the motion $x(\cdot)=x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$ of system Eq. 7, the inequality below holds:

[TABLE]

• $iii)$

there exists $L_{x}>0$ such that, for any $(t,w(\cdot))$, $(t,w^{\prime}(\cdot))\in G$ satisfying $\|w(\cdot)\|_{[0,t]}\leq R$ and $\|w^{\prime}(\cdot)\|_{[0,t]}\leq R$, any $\theta\in[t,T]$, and any $u(\cdot)\in\mathcal{U}(t,\theta)$, for the motions $x(\cdot)=x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$ and $x^{\prime}(\cdot)=x(\cdot\mid t,w^{\prime}(\cdot),\theta,u(\cdot))$ of system Eq. 7, the following inequality is valid:

[TABLE]

The proof of this proposition is based on the following two lemmas.

Lemma 7.2**.**

For any $(t,w(\cdot))\in G$, the inequality below holds:

[TABLE]

This lemma is proved by the scheme from [38, Theorem 14.10 and Corollary 1].

Lemma 7.3** ([10, Lemma 6.19]).**

Let $t\in[0,T]$, $\theta\in[t,T]$, $\psi(\cdot)\in{C}([t,\theta],\mathbb{R})$, $a\geq 0$, and $b\geq 0$ be such that

[TABLE]

Then, the following inequalities are valid:

[TABLE]

*where $E_{\alpha}$ is the Mittag-Leffler function of the order $\alpha$ (see, e.g., [38, (1.90)]). *

Proof 7.4** (Proof of Proposition 7.1).**

Let $R>0$ be fixed.

Let us prove part $i)$. Taking $c_{f}$ from Eq. 9, we define $M_{x}=(1+3R)E_{\alpha}(T^{\alpha}c_{f})-1$. Let $(t,w(\cdot))\in G$ such that $\|w(\cdot)\|_{[0,t]}\leq R$, $\theta\in[t,T]$, and $u(\cdot)\in\mathcal{U}(t,\theta)$ be fixed, and let $x(\cdot)=x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$ be the corresponding motion of system Eq. 7. Due to Eqs. 25 and 7.2, we derive

[TABLE]

Hence,

[TABLE]

wherefrom, applying Lemma 7.3, we obtain $\|x(\tau)\|\leq M_{x}$ for $\tau\in[t,\theta]$. Since $M_{x}\geq R$, then this estimate also holds for $\tau\in[0,t)$. Thus, $\|x(\cdot)\|_{[0,\theta]}\leq M_{x}$.

Further, we prove part $ii)$. Let $H_{x}=H_{\alpha}\max\{R,(1+M_{x})c_{f}\}$, where $H_{\alpha}$ is taken from Eq. 1. Let $(t,w(\cdot))\in G$ such that $\|w(\cdot)\|_{[0,t]}\leq R$ and $\|(^{C}D^{\alpha}w)(\cdot)\|_{[0,t]}\leq R$, $\theta\in[t,T]$, and $u(\cdot)\in\mathcal{U}(t,\theta)$ be fixed, and let $x(\cdot)=x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$. We have

[TABLE]

Moreover, due to Eqs. 7 and 9, we obtain

[TABLE]

Then, $\|(^{C}D^{\alpha}x)(\cdot)\|_{[0,\theta]}\leq\max\{R,(1+M_{x})c_{f}\}$, and Eq. 32 follows from Eqs. 5 and 1.

Finally, let us prove part $iii)$. According to local Lipschitz continuity of $f$ in $x$ (see Section 3), let us choose $\lambda_{f}$ by $M_{x}$ and define $L_{x}=3E_{\alpha}(T^{\alpha}\lambda_{f})$. Let $(t,w(\cdot)),$ $(t,w^{\prime}(\cdot))\in G$ such that $\|w(\cdot)\|_{[0,t]}\leq R$ and $\|w^{\prime}(\cdot)\|_{[0,t]}\leq R$, $\theta\in[t,T]$, and $u(\cdot)\in\mathcal{U}(t,\theta)$ be fixed. Let us consider the motions $x(\cdot)=x(\cdot\mid t,w(\cdot),\theta,u(\cdot))$ and $x^{\prime}(\cdot)=x(\cdot\mid t,w^{\prime}(\cdot),\theta,u(\cdot))$ of system Eq. 7. Due to Eq. 25, we derive

[TABLE]

Let us denote $\bar{w}(\tau)=w(\tau)-w^{\prime}(\tau)$, $\tau\in[0,t]$. Then, $\bar{w}(\cdot)\in{AC^{\alpha}}([0,t],\mathbb{R}^{n})$ and, consequently, $(t,\bar{w}(\cdot))\in G$. Moreover, in accordance with Eq. 4, we have $({}^{C}D^{\alpha}\bar{w})(\tau)=({}^{C}D^{\alpha}w)(\tau)-({}^{C}D^{\alpha}w^{\prime})(\tau)$ for a.e. $\tau\in[0,t]$. Therefore, applying Lemma 7.2, we get

[TABLE]

where $\tau\in[t,\theta]$. Hence,

[TABLE]

*wherefrom, by Lemma 7.3, we obtain $\|x(\tau)-x^{\prime}(\tau)\|\leq L_{x}\|w(\cdot)-w^{\prime}(\cdot)\|_{[0,t]}$ for $\tau\in[t,\theta]$. Since $L_{x}\geq 1$, then this estimate also holds for $\tau\in[0,t)$, and Eq. 33 is proved. *

Let us emphasize that, in part $ii)$ of Proposition 7.1, the Hölder constant $H_{x}$ of the motion $x(\cdot)$ depends not only on the estimate of the history $w(\cdot)$, but also on the estimate of its derivative $(^{C}D^{\alpha}w)(\cdot)$.

8 Continuity of the value functional

First, let us prove the uniform continuity of the value functional $\rho:G\rightarrow\mathbb{R}$ (see Eq. 28) in the functional argument $w(\cdot)$.

Lemma 8.1**.**

For any $R>0$ and $\varepsilon>0$, there exists $\delta>0$ such that, for any $(t,w(\cdot))$, $(t,w^{\prime}(\cdot))\in G$, if $\|w(\cdot)\|_{[0,t]}\leq R$ and $\|w(\cdot)-w^{\prime}(\cdot)\|_{[0,t]}\leq\delta$, then

[TABLE]

Proof 8.2**.**

Let $R>0$ and $\varepsilon>0$ be fixed. By $R^{\prime}=R+1$, let us choose $M_{x}^{\prime}$ and $L_{x}^{\prime}$ according to Proposition 7.1. Due to continuity of $\sigma$ and $\chi$ (see Section 3), there exists $\delta\in(0,1]$ such that, for any $x,$ $x^{\prime}\in B(M_{x}^{\prime})$ satisfying $\|x-x^{\prime}\|\leq L_{x}^{\prime}\delta$, any $\tau\in[0,T]$, and any $u\in P$, the following inequalities are valid:

[TABLE]

Let us show that the statement of the lemma is valid for this $\delta$.

Let $(t,w(\cdot)),$ $(t,w^{\prime}(\cdot))\in G$ be fixed, and let $\|w(\cdot)\|_{[0,t]}\leq R$, $\|w(\cdot)-w^{\prime}(\cdot)\|_{[0,t]}\leq\delta$. Let us note that $\|w^{\prime}(\cdot)\|_{[0,t]}\leq R^{\prime}$. For every $u(\cdot)\in\mathcal{U}(t,T)$, we consider the motions $x(\cdot)=x(\cdot\mid t,w(\cdot),T,u(\cdot))$ and $x^{\prime}(\cdot)=x(\cdot\mid t,w^{\prime}(\cdot),T,u(\cdot))$ of system Eq. 7. We have $\|x(\cdot)\|_{[0,T]}\leq M_{x}^{\prime}$, $\|x^{\prime}(\cdot)\|_{[0,T]}\leq M_{x}^{\prime}$, and $\|x(\cdot)-x^{\prime}(\cdot)\|_{[0,T]}\leq L_{x}^{\prime}\delta$. Hence,

[TABLE]

*Since this estimate is valid for every $u(\cdot)\in\mathcal{U}(t,T)$, then, in accordance with Eq. 28, we conclude Eq. 34. The lemma is proved. *

In order to study the continuity of $\rho$ with respect to both variables $t$ and $w(\cdot)$, the set $G$ is endowed with the following metric (see, e.g., [31, p. 25] and also [30]):

[TABLE]

where $(t,w(\cdot))$, $(t^{\prime},w^{\prime}(\cdot))\in G$,

[TABLE]

and $\operatorname{dist}^{\ast}((t^{\prime},w^{\prime}(\cdot)),(t,w(\cdot)))$ is defined in a similar way with clear changes. Let us note that this metric is a Hausdorff distance between the graphics of the functions $w:[0,t]\rightarrow\mathbb{R}^{n}$ and $w^{\prime}:[0,t^{\prime}]\rightarrow\mathbb{R}^{n}$.

The proposition below holds (see also [31, (P2.12)]).

Proposition 8.3**.**

For any $(t,w(\cdot))$, $(t^{\prime},w^{\prime}(\cdot))\in G$, $t^{\prime}\leq t$, the inequalities

[TABLE]

*are valid, where $\operatorname{dist}=\operatorname{dist}((t,w(\cdot)),(t^{\prime},w^{\prime}(\cdot)))$, $\varkappa$ is the modulus of continuity of $w(\cdot)$, and $w_{t^{\prime}}(\tau)=w(\tau)$, $\tau\in[0,t^{\prime}]$. *

Proof 8.4**.**

Let $(t,w(\cdot))$, $(t^{\prime},w^{\prime}(\cdot))\in G$ such that $t^{\prime}\leq t$ be fixed.

For every $\tau\in[0,t]$, if $\tau\leq t^{\prime}$, then

[TABLE]

and, if $\tau>t^{\prime}$, then

[TABLE]

Therefore, we obtain

[TABLE]

On the other hand, for every $\tau^{\prime}\in[0,t^{\prime}]$, we have

[TABLE]

Thus,

[TABLE]

and we conclude Eq. 36.

Further, the first inequality in Eq. 37 follows from the estimates

[TABLE]

Let us prove the second inequality in Eq. 37. Let us choose $\tau^{\prime}_{0}\in[0,t^{\prime}]$ from the condition $\|w_{t^{\prime}}(\cdot)-w^{\prime}(\cdot)\|_{[0,t^{\prime}]}=\|w(\tau^{\prime}_{0})-w^{\prime}(\tau^{\prime}_{0})\|$ and take $\tau_{0}\in[0,t]$ such that

[TABLE]

Hence, since

[TABLE]

then $|\tau_{0}-\tau^{\prime}_{0}|\leq\operatorname{dist}$, $\|w(\tau_{0})-w^{\prime}(\tau^{\prime}_{0})\|\leq\operatorname{dist}$, and, consequently,

[TABLE]

*Therefore, due to the choice of $\tau^{\prime}_{0}$, we get the second inequality in Eq. 37. *

Now, let us prove that the value functional $\rho:G\rightarrow\mathbb{R}$ (see Eq. 28) is continuous.

Theorem 8.5**.**

For any $(t,w(\cdot))\in G$ and $\varepsilon>0$, there exists $\delta>0$ such that, for any $(t^{\prime},w^{\prime}(\cdot))\in G$, from the inequality $\operatorname{dist}((t,w(\cdot)),(t^{\prime},w^{\prime}(\cdot)))\leq\delta$, it follows that

[TABLE]

Proof 8.6**.**

Let $(t,w(\cdot))\in G$ and $\varepsilon>0$ be fixed. Let $R>0$ be such that $\|w(\cdot)\|_{[0,t]}\leq R$ and $\|(^{C}D^{\alpha}w)(\cdot)\|_{[0,t]}\leq R$, and let $M_{x}$ and $H_{x}$ be chosen according to Proposition 7.1. Due to continuity of $\chi$ (see Section 3), there exists $M_{\chi}>0$ such that

[TABLE]

Let $\delta_{1}>0$ be such that $M_{\chi}\delta_{1}\leq\varepsilon/6$. Further, applying Lemma 8.1 for $M_{x}$ and $\varepsilon/3$, let us choose $\delta_{2}$. Finally, taking $c_{f}$ from Eq. 9, we consider $\delta_{3}>0$ such that

[TABLE]

Let us show that the statement of the theorem is valid for $\delta=\min\{\delta_{1},\delta_{3}\}$.

Let $(t^{\prime},w^{\prime}(\cdot))\in G$ be fixed such that $\operatorname{dist}((t,w(\cdot)),(t^{\prime},w^{\prime}(\cdot)))\leq\delta$. Let us note that $|t-t^{\prime}|\leq\delta$ due to Proposition 8.3.

Let us consider the case when $t^{\prime}\leq t$. Taking into account that $w(\cdot)$ is Hölder continuous of the order $\alpha$ with the constant $H_{x}$, applying Proposition 8.3, we derive $\|w_{t^{\prime}}(\cdot)-w^{\prime}(\cdot)\|_{[0,t^{\prime}]}\leq\delta+H_{x}\delta^{\alpha}\leq\delta_{2}$. Since $\|w_{t^{\prime}}(\cdot)\|_{[0,t^{\prime}]}\leq M_{x}$, then, by Lemma 8.1,

[TABLE]

According to Theorem 6.1, there exists $u^{\prime}(\cdot)\in\mathcal{U}(t^{\prime},t)$ such that, for the corrsponding motion $x^{\prime}(\cdot)=x(\cdot\mid t^{\prime},w_{t^{\prime}}(\cdot),t,u^{\prime}(\cdot))$ of system Eq. 7, we have

[TABLE]

Hence, taking into account that $\|x^{\prime}(\cdot)\|_{[0,t]}\leq M_{x}$, we get

[TABLE]

Due to Eqs. 25 and 5, applying Eq. 9, we derive

[TABLE]

Thus, since $x^{\prime}(\tau)=w(\tau)$, $\tau\in[0,t^{\prime}]$, then $\|x^{\prime}(\cdot)-w(\cdot)\|_{[0,t]}\leq\delta_{2}$, and, by Lemma 8.1,

[TABLE]

From Eqs. 40, 41, and 42, we conclude Eq. 38.

Now, let us suppose that $t^{\prime}>t$. By Theorem 6.1, there exists $u(\cdot)\in\mathcal{U}(t,t^{\prime})$ such that, for the motion $x(\cdot)=x(\cdot\mid t,w(\cdot),t^{\prime},u(\cdot))$ of system Eq. 7, we have

[TABLE]

and, consequently, taking into account that $\|x(\cdot)\|_{[0,t^{\prime}]}\leq M_{x}$, we get

[TABLE]

Since $x(\cdot)$ is Hölder continuous of the order $\alpha$ with the constant $H_{x}$ and $x(\tau)=w(\tau)$, $\tau\in[0,t]$, then, applying Proposition 8.3, we derive $\operatorname{dist}((t^{\prime},x(\cdot)),(t,w(\cdot)))\leq\delta+H_{x}\delta^{\alpha}$. Hence, by the triangle inequality,

[TABLE]

Thus, using Proposition 8.3 again, we obtain

[TABLE]

Therefore, according to Lemma 8.1, we conclude

[TABLE]

*From Eqs. 43 and 44, we derive Eq. 38. The theorem is proved. *

9 Coinvariant derivatives of fractional order

In order to write out the Hamilton–Jacobi–Bellman equation associated with the considered optimal control problem Eqs. 7 and 11, let us introduce an appropriate notion of derivatives of functionals defined on the set $G$ of positions of system Eq. 7.

Let us denote $G^{0}=\{(t,w(\cdot))\in G:\,t<T\}$, and, for every $(t,w(\cdot))\in G^{0}$, in accordance with Eq. 24, consider the set of admissible extensions $x(\cdot)$ of $w(\cdot)$:

[TABLE]

A functional $\varphi:G\rightarrow\mathbb{R}$ is called coinvariantly ($ci$-) differentiable of the order $\alpha$ at $(t,w(\cdot))\in G^{0}$, if there exist $\partial_{t}^{\alpha}\varphi(t,w(\cdot))\in\mathbb{R}$ and $\nabla^{\alpha}\varphi(t,w(\cdot))\in\mathbb{R}^{n}$ such that, for every $x(\cdot)\in X(t,w(\cdot))$ and $\tau\in(t,T)$, the relation below is valid:

[TABLE]

where $x_{\tau}(\cdot)$ is determined by $x(\cdot)$ according to Eq. 26, $o(\tau-t)$ may depend on $x(\cdot)$, and $o(\tau-t)/(\tau-t)\rightarrow 0$ when $\tau\downarrow t$. Respectively, $\partial_{t}^{\alpha}\varphi(t,w(\cdot))$ and $\nabla^{\alpha}\varphi(t,w(\cdot))$ are called the $ci$-derivative in $t$ and the $ci$-gradient of the order $\alpha$ of $\varphi$ at $(t,w(\cdot))$. Let us note that, due to Eqs. 6 and 45, relation Eq. 46 can be rewritten as follows:

[TABLE]

Remark 9.1**.**

In the case $\alpha=1$, the introduced notion of $ci$-differentiability of the order $\alpha$ agrees with the notion of $ci$-differentiability developed, e.g., in [31, § 2] and [25, § 2.4] (see also [24, 29, 30]). Indeed, in this case, the set $X(t,w(\cdot))$ consists of all Lipschitz continuous extensions of $w(\cdot)$, and Eq. 46 becomes

[TABLE]

Let us note that Eq. 46 determines the derivatives $\partial_{t}^{\alpha}\varphi(t,w(\cdot))$ and $\nabla^{\alpha}\varphi(t,w(\cdot))$ uniquely. In order to verify this, one should take an arbitrary $l\in\mathbb{R}^{n}$, consider the extension $x^{(l)}(\cdot)\in X(t,w(\cdot))$ such that $(^{C}D^{\alpha}x)(\tau)=l$, $\tau\in(t,T]$, substitute it in Eq. 46 (or, equivalently, in Eq. 47), and take into account the invariance of $\partial_{t}^{\alpha}\varphi(t,w(\cdot))$ and $\nabla^{\alpha}\varphi(t,w(\cdot))$ with respect to $x^{(l)}(\cdot)$ (see also the arguments in [31, p. 34]).

Further, a functional $\varphi:G\rightarrow\mathbb{R}$ is called $ci$-smooth of the order $\alpha$, if it satisfies the following conditions: (a) $\varphi$ is continuous; (b) $\varphi$ is $ci$-differentiable of the order $\alpha$ at every $(t,w(\cdot))\in G^{0}$; (c) the functionals $\partial_{t}^{\alpha}\varphi:G^{0}\rightarrow\mathbb{R}$ and $\nabla^{\alpha}\varphi:G^{0}\rightarrow\mathbb{R}^{n}$ are continuous. We recall that the set $G^{0}\subset G$ is endowed with the metric $\operatorname{dist}$ (see Eq. 35).

The notion of fractional $ci$-derivatives allows us to obtain a simple formula for the total derivative of a $ci$-smooth functional along motions of system Eq. 7.

Lemma 9.2**.**

Let a functional $\varphi:G\rightarrow\mathbb{R}$ be $ci$-smooth of the order $\alpha$. Then, for any $(t,w(\cdot))\in G^{0}$, $x(\cdot)\in X(t,w(\cdot))$, and $\theta\in[t,T)$, the function $\omega(\tau)=\varphi(\tau,x_{\tau}(\cdot))$, $\tau\in[t,T]$, is continuous on $[t,T]$ and Lipschitz continuous on $[t,\theta]$. Moreover,

[TABLE]

Proof 9.3**.**

We follow the scheme from [31, Lemma 2.1] (see also [35, Lemma 3]).

Let $(t,w(\cdot))\in G^{0}$, $x(\cdot)\in X(t,w(\cdot)$, and $\theta\in[t,T)$ be fixed. Since $x(\cdot)$ is continuous, then, by Proposition 8.3, the function $[t,T]\ni\tau\mapsto(\tau,x_{\tau}(\cdot))\in G$ is continuous. Hence, due to continuity of $\varphi$, we obtain that $\omega(\cdot)$ is continuous on $[t,T]$. Moreover, since the functionals $\partial_{t}^{\alpha}\varphi$ and $\nabla^{\alpha}\varphi$ are also continuous, there exists $M>0$ such that $|\partial_{t}^{\alpha}\varphi(\tau,x_{\tau}(\cdot))|\leq M$ and $\|\nabla^{\alpha}\varphi(\tau,x_{\tau}(\cdot))\|\leq M$ for $\tau\in[t,\theta]$. Let us define $L=(1+\|(^{C}D^{\alpha}x)(\cdot)\|_{[t,T]})M$ and prove the inequality

[TABLE]

Let $\tau\in[t,\theta]$ be fixed. Since $\varphi$ is $ci$-differentiable of the order $\alpha$ at $(\tau,x_{\tau}(\cdot))$ and $x(\cdot)\in X(\tau,x_{\tau}(\cdot))$, then, according to Eq. 47, for every $\delta\in(0,T-\tau)$, we have

[TABLE]

and, hence, $|\omega(\tau+\delta)-\omega(\tau)|/\delta\leq L+|o(\delta)|/\delta$. Thus, we get

[TABLE]

Taking into account that this inequality hods for every $\tau\in[t,\theta]$, and $\omega(\cdot)$ is continuous on $[t,\theta]$, by Dini’s theorem (see, e.g., [7, Ch. 4, Theorem 1.2]), we conclude Eq. 49.

Further, let us prove Eq. 48. Namely, let us show that the equality in Eq. 48 is valid for every $\tau\in(t,\theta)$ such that the derivatives $(^{C}D^{\alpha}x)(\tau)$ and $\mathrm{d}\omega(\tau)/\mathrm{d}\tau$ exist. According to Eqs. 46 and 4, we derive

[TABLE]

*The lemma is proved. *

Remark 9.4**.**

*It seems that the proposed notion of fractional $ci$-differentiation may also be used for the needs of the stability theory for fractional-order systems. Namely, the absence of a simple rule for calculating the fractional derivative of the composition of two functions (see, e.g., [40] and the references therein) leads to the difficulties in defining the notion of the total derivative of a Lyapunov function $V(t,x)$ along motions of the system (see, e.g. [1]). In particular, it becomes important and relevant to obtain different estimates for the fractional derivative of the composition $V(t,x(t))$ of the function $V(t,x)$ and a motion $x(\cdot)$ (see, e.g., [3, 2, 14, 43] and the references therein). However, according to the discussion in Section 4 (see also, e.g., [8]), it seems that, by analogy with functional-differential systems (see [27] and also [25]), for studying the questions of stability for fractional-order systems, it may be useful to apply Lyapunov–Krasovskii functionals $V(t,w(\cdot))$ that depend on the history $w(\cdot)$ of a motion $x(\cdot)$. In this case, Lemma 9.2 gives a simple formula for the total derivative of $V(t,w(\cdot))$ along motions of the system. *

10 Hamilton–Jacobi–Bellman equation

With the optimal control problem Eqs. 7 and 11, let us associate the Hamilton–Jacobi–Bellman equation

[TABLE]

where $\partial_{t}^{\alpha}\varphi(t,w(\cdot))$ and $\nabla^{\alpha}\varphi(t,w(\cdot))$ are the $ci$-derivatives of the order $\alpha$ of $\varphi$ at $(t,w(\cdot))$ (see Section 9), and the Hamiltonian is defined in the usual way:

[TABLE]

The main result of this section is the following.

Theorem 10.1**.**

*If the value functional $\rho$ (see Eq. 28) is $ci$-smooth, then it satisfies Hamilton–Jacobi–Bellman equation Eq. 50. *

Let us note that, since $\rho$ is always continuous by Theorem 8.5, then the essential assumptions of Theorem 10.1 are $ci$-differentiability of the order $\alpha$ of $\rho$ at every $(t,w(\cdot))\in G^{0}$ and continuity of the functionals $\partial_{t}^{\alpha}\rho:G^{0}\rightarrow\mathbb{R}$ and $\nabla^{\alpha}\rho:G^{0}\rightarrow\mathbb{R}^{n}$.

In order to prove Theorem 10.1, and also Theorem 11.1 below, we need some auxiliary constructions.

For every $(t,w(\cdot))\in G^{0}$, taking $c_{f}$ from Eq. 9, let us consider the sets

[TABLE]

where $X(t,w(\cdot))$ is defined according to Eq. 45, and

[TABLE]

In the proposition below, some properties of these sets are given.

Proposition 10.2**.**

For any $(t,w(\cdot))\in G^{0}$, the following statements are valid:

• $i)$

for any $u(\cdot)\in\mathcal{U}(t,T)$, the motion $x(\cdot)=x(\cdot\mid t,w(\cdot),T,u(\cdot))$ of system Eq. 7 satisfies the inclusion $x(\cdot)\in X_{\ast}(t,w(\cdot))$; therefore, in particular, we have $(\tau,x_{\tau}(\cdot))\in G_{\ast}(t,w(\cdot))$, $\tau\in[0,T]$;

• $ii)$

there exist $R_{\ast}>0$ and $H_{\ast}>0$ such that, for any $x(\cdot)\in X_{\ast}(t,w(\cdot)),$ the inequalities below hold:

[TABLE]

• $iii)$

the set $X_{\ast}(t,w(\cdot))$ is a compact subset of ${AC^{\alpha}}([0,T],\mathbb{R}^{n})$;

• $iv)$

the set $G_{\ast}(t,w(\cdot))$ is a compact subset of $G$.

Proof 10.3**.**

Part $i)$ follows from definition Eq. 52 of $X_{\ast}(t,w(\cdot))$ and the sublinear growth of $f$ in $x$ (see Section 3). The proof of $ii)$ is carried out by analogy with parts $i)$ and $ii)$ of Proposition 7.1. Part $iii)$ is a consequence of [13, Assertion 7].

*Let us prove part $iv)$. Let $(t,w(\cdot))\in G^{0}$ and a sequence $\{(t_{m},w_{m}(\cdot))\}_{m\in\mathbb{N}}\subset G_{\ast}(t,w(\cdot))$ be fixed. For every $m\in\mathbb{N}$, let us choose $x^{(m)}(\cdot)\in X_{\ast}(t,w(\cdot))$ such that $x_{t_{m}}^{(m)}(\cdot)=w_{m}(\cdot)$. According to part $iii)$, there exist $\theta\in[0,T]$, $x_{\ast}(\cdot)\in X_{\ast}(t,w(\cdot))$ and subsequences $\{t_{m_{i}}\}_{i\in\mathbb{N}}$, $\{x^{(m_{i})}(\cdot)\}_{i\in\mathbb{N}}$ such that $t_{m_{i}}\rightarrow\theta$ and $\|x_{\ast}(\cdot)-x^{(m_{i})}(\cdot)\|_{[0,T]}\rightarrow 0$ when $i\rightarrow\infty$. Let us consider the function $w_{\ast}(\tau)=x_{\ast}(\tau),$ $\tau\in[0,\theta]$. The inclusion $x_{\ast}(\cdot)\in X_{\ast}(t,w(\cdot))$ implies $(\theta,w_{\ast}(\cdot))\in G_{\ast}(t,w(\cdot))$. Applying Proposition 8.3 (see also [31, Lemma 1.1]), we derive $\operatorname{dist}((\theta,w_{\ast}(\cdot)),(t_{m_{i}},w_{m_{i}}(\cdot)))\rightarrow 0$ when $i\rightarrow\infty$, and, therefore, the subsequence $\{(t_{m_{i}},w_{m_{i}}(\cdot))\}_{i\in\mathbb{N}}$ converges to $(\theta,w_{\ast}(\cdot))\in G_{\ast}(t,w(\cdot))$. Thus, the set $G_{\ast}(t,w(\cdot))$ is a compact subset of $G$. The proposition is proved. *

Proof 10.4** (Proof of Theorem 10.1).**

Let us note that Lemma 9.2 allows us to prove the theorem by applying the standard arguments (see, e.g., [44, Proposition 2.4.3]).

Let $(t,w(\cdot))\in G^{0}$ be fixed, and let $R_{\ast}$ and $H_{\ast}$ be chosen according to Proposition 10.2. Let us take $\theta\in(t,T)$ and introduce the set

[TABLE]

It follows from part $iv)$ of Proposition 10.2 that $G^{\theta}_{\ast}(t,w(\cdot))$ is a compact subset of $G^{0}$. Therefore, since the functional $\nabla^{\alpha}\rho:G^{0}\rightarrow\mathbb{R}^{n}$ is continuous, there exists $M>0$ such that $\|\nabla^{\alpha}\rho(t^{\prime},w^{\prime}(\cdot))\|\leq M$ for $(t^{\prime},w^{\prime}(\cdot))\in G^{\theta}_{\ast}(t,w(\cdot))$.

Further, let us fix $\varepsilon>0$. Due to continuity of the functionals $\partial_{t}^{\alpha}\rho:G^{0}\rightarrow\mathbb{R}$ and $\nabla^{\alpha}\rho$, let us choose $\delta_{1}>0$ such that, for any $(t^{\prime},w^{\prime}(\cdot))$, $(t^{\prime\prime},w^{\prime\prime}(\cdot))\in G^{\theta}_{\ast}(t,w(\cdot))$ satisfying $\operatorname{dist}((t^{\prime},w^{\prime}(\cdot)),(t^{\prime\prime},w^{\prime\prime}(\cdot)))\leq\delta_{1}+H_{\ast}\delta_{1}^{\alpha}$, the inequality below holds:

[TABLE]

where $c_{f}$ is taken from Eq. 9. Moreover, since $f$ and $\chi$ are continuous (see Sections 3 and 3), there exists $\delta_{2}>0$ such that, for any $\tau,$ $\tau^{\prime}\in[0,T]$, $x$, $x^{\prime}\in B(R_{\ast})$ satisfying $|\tau-\tau^{\prime}|\leq\delta_{2}$, $\|x-x^{\prime}\|\leq H_{\ast}\delta_{2}^{\alpha}$, and any $u\in P$, we get

[TABLE]

Let us define $\delta=\min\{\delta_{1},\delta_{2},\theta-t\}$. Hence, according to Propositions 8.3 and 10.2, we obtain the following property. Let $u(\cdot)\in\mathcal{U}(t,T)$, and let $x(\cdot)=x(\cdot\mid t,w(\cdot),T,u(\cdot))$ be the motion of system Eq. 7. Then, for every $u^{\prime}\in P$, denoting

[TABLE]

for $\tau\in[t,\theta]$, we have $|\mu(\tau)-\mu(\tau^{\prime})|\leq\varepsilon$ for any $\tau,$ $\tau^{\prime}\in[t,\theta]$ such that $|\tau-\tau^{\prime}|\leq\delta$.

Now, let us prove that

[TABLE]

For every $u\in P$, let us consider the constant control $u(\tau)=u$, $\tau\in[t,t+\delta]$, and the corresponding motion $x(\cdot)=x(\cdot\mid t,w(\cdot),t+\delta,u(\cdot))$ of system Eq. 7. By Theorem 6.1,

[TABLE]

Formally extending the motion $x(\cdot)$ up to $T$ and applying Lemma 9.2, we obtain that the function $\omega(\tau)=\rho(\tau,x_{\tau}(\cdot)),$ $\tau\in[t,t+\delta]$, is Lipschitz continuous and

[TABLE]

Therefore, we have

[TABLE]

From Eqs. 55 and 56, we derive

[TABLE]

Hence, the choice of $\delta$ yields (see Eq. 53)

[TABLE]

Since this estimate holds for every $u\in P$ and $\varepsilon>0$, then, due to Eq. 51, we get Eq. 54.

On the other hand, let us show that

[TABLE]

According to Theorem 6.1, there exists $u(\cdot)\in\mathcal{U}(t,t+\delta)$ such that

[TABLE]

where $x(\cdot)=x(\cdot\mid t,w(\cdot),t+\delta,u(\cdot))$. Then, arguing as above, we obtain

[TABLE]

and, by the choice of $\delta$,

[TABLE]

Hence, taking Eq. 51 into account, we derive

[TABLE]

Since this inequality is valid for every $\varepsilon>0$, then we get Eq. 57.

*From Eqs. 54 and 57, we conclude Eq. 50. The theorem is proved. *

11 Optimal control strategy

In this section, we establish the result that, in some sense, is converse to Theorem 10.1. Namely, we prove that a $ci$-smooth of the order $\alpha$ functional $\varphi:G\rightarrow\mathbb{R}$ that satisfies Hamilton–Jacobi–Bellman equation Eq. 50 and the natural right-end condition (see Eq. 30)

[TABLE]

coincides with the value functional $\rho$ (see Eq. 28). Furthermore, we propose a way of forming $\varepsilon$-optimal controls on the basis of this functional $\varphi$ by using a stepwise feedback control scheme. In order to formulate this result, we consider the following formalization of such control schemes, which goes back to the positional approach in differential games [28, 26] (see also [33, 31, 29, 30] and [13, 15]).

By a control strategy, we mean an arbitrary functional $U:G^{0}\rightarrow P$. Let us fix a position $(t,w(\cdot))\in G^{0}$ and a partition of the time interval $[t,T]$:

[TABLE]

The pair $\{U,\Delta\}$ is called a control law. This control law forms a piecewise constant control $u(\cdot)\in\mathcal{U}(t,T)$ and the corresponding motion $x(\cdot)=x(\cdot\mid t,w(\cdot),T,u(\cdot))$ of system Eq. 7 according to the following recursive procedure: at every time $\tau_{j}$, $j\in\overline{1,k}$, the history $x_{\tau_{j}}(\cdot)$ (see Eq. 26) of the motion $x(\cdot)$ on $[0,\tau_{j}]$ is measured, the value $u_{j}=U(\tau_{j},x_{\tau_{j}}(\cdot))$ is computed, and then the constant control $u(\tau)=u_{j}$ is applied until $\tau_{j+1}$, when a new measurement of the history is taken. In a short form, we have

[TABLE]

Formally putting $u(T)=\tilde{u}$ for some fixed $\tilde{u}\in P$, we conclude that the described procedure determines $u(\cdot)$ and $x(\cdot)$ uniquely. For the obtained control $u(\cdot)$, we also use the notation $u(\cdot\mid t,w(\cdot),U,\Delta)$. Let us note that, according to Eq. 28, the corresponding value of cost functional Eq. 27 satisfies the estimate

[TABLE]

Taking this into account, we call a control strategy $U^{\circ}$ optimal if the following statement holds. For any $(t,w(\cdot))\in G^{0}$ and $\varepsilon>0$, there exists $\delta>0$ such that, for any partition $\Delta$ (see Eq. 59) with the diameter $\operatorname{diam}(\Delta)=\max_{j\in\overline{1,k}}(\tau_{j+1}-\tau_{j})\leq\delta$, the inequality below is valid:

[TABLE]

or, in other words, the control $u(\cdot\mid t,w(\cdot),U^{\circ},\Delta)$ is $\varepsilon$-optimal. Thus, the problem of constructing $\varepsilon$-optimal controls can be reduced to finding an optimal control strategy $U^{\circ}$.

The main result of this section is the following.

Theorem 11.1**.**

Let a $ci$-smooth of the order $\alpha$ functional $\varphi:G\rightarrow\mathbb{R}$ satisfies Hamilton–Jacobi–Bellman equation Eq. 50 and right-end condition Eq. 58. Then,

[TABLE]

and the control strategy

[TABLE]

*where $(t,w(\cdot))\in G^{0}$, is optimal. *

The proof of the theorem follows the scheme from, e.g., [31, Theorem 3.1] (see also [30, Theorem 3.1] and [16, Theorem 1]), and, for convenience, is divided into the following two lemmas, which are valid under the assumptions of the theorem.

Lemma 11.2**.**

For any $(t,w(\cdot))\in G^{0}$, the inequality below holds:

[TABLE]

Proof 11.3**.**

Let us fix $(t,w(\cdot))\in G^{0}$, take $u(\cdot)\in\mathcal{U}(t,T)$, and show that

[TABLE]

Let us consider the function

[TABLE]

where $x(\cdot)=x(\cdot\mid t,w(\cdot),T,u(\cdot))$ is the motion of system Eq. 7. Let $\varepsilon>0$ be fixed. Due to Lemma 9.2 and continuity of $\chi$, the function $\omega(\cdot)$ is continuous, and, therefore, one can choose $\theta\in(t,T)$ such that

[TABLE]

Moreover, the function $\omega(\cdot)$ is Lipschitz continuous on $[t,\theta]$ and, for a.e. $\tau\in[t,\theta]$,

[TABLE]

Further, according to Eqs. 51 and 50, we have

[TABLE]

Consequently, we derive $\mathrm{d}\omega(t)/\mathrm{d}t\geq 0$ for a.e. $\tau\in[t,\theta]$, and, hence, $\omega(\theta)-\omega(t)\geq 0$. Thus, it follows from Eq. 66 that $\omega(T)-\omega(t)\geq-\varepsilon$. Since this inequality holds for every $\varepsilon>0$, then $\omega(T)-\omega(t)\geq 0$. Taking into account that, due to Eqs. 58 and 65,

[TABLE]

the obtained estimate yields Eq. 64.

*Since Eq. 64 is valid for every $u(\cdot)\in\mathcal{U}(t,T)$, then, in accordance with Eq. 28, we get Eq. 63. The lemma is proved. *

Lemma 11.4**.**

For any $(t,w(\cdot))\in G^{0}$ and $\varepsilon>0$, there exists $\delta>0$ such that, for any partition $\Delta$ (see Eq. 59) satisfying $\operatorname{diam}(\Delta)\leq\delta$, the inequality below holds:

[TABLE]

Proof 11.5**.**

Let $(t,w(\cdot))\in G^{0}$ and $\varepsilon>0$ be fixed. Let $R_{\ast}$ and $H_{\ast}$ be taken from Proposition 10.2. Due to continuity of $\varphi:G\rightarrow\mathbb{R}$ and compactness of $G_{\ast}(t,w(\cdot))$ (see Proposition 10.2), let us choose $\eta_{1}>0$ such that, for any $(t^{\prime},w^{\prime}(\cdot))$, $(t^{\prime\prime},w^{\prime\prime}(\cdot))\in G_{\ast}(t,w(\cdot))$ satisfying $\operatorname{dist}((t^{\prime},w^{\prime}(\cdot)),(t^{\prime\prime},w^{\prime\prime}(\cdot)))\leq\eta_{1}+H_{\ast}\eta_{1}^{\alpha}$, we have

[TABLE]

Since $\chi$ is continuous, then there exists $M_{\chi}>0$ such that Eq. 39 is valid where $R_{\ast}$ is substituted instead of $M_{x}$. Let us choose $\eta_{2}>0$ from the condition $M_{\chi}\eta_{2}\leq\varepsilon/4$ and take $\theta\in(t,T)$ such that $T-\theta\leq\min\{\eta_{1},\eta_{2}\}$. Then, according to Propositions 8.3 and 10.2, we obtain the following property. Let $u(\cdot)\in\mathcal{U}(t,T)$, and let $x(\cdot)=x(\cdot\mid t,w(\cdot),T,u(\cdot))$ be the corresponding motion of system Eq. 7. Then,

[TABLE]

Further, arguing as in the proof of Theorem 10.1, let us choose $\delta>0$ such that, if $u(\cdot)\in\mathcal{U}(t,T)$, $x(\cdot)=x(\cdot\mid t,w(\cdot),T,u(\cdot))$, and $u^{\prime}\in P$, then the function $\mu(\cdot)$ defined by Eq. 53 where the functional $\varphi$ is substituted instead of $\rho$ satisfies the inequality $|\mu(\tau)-\mu(\tau^{\prime})|\leq\varepsilon/(2(\theta-t))$ for any $\tau,$ $\tau^{\prime}\in[t,\theta]$ satisfying $|\tau-\tau^{\prime}|\leq\delta$. Let us show that the statement of the lemma is valid for the chosen $\delta$.

Let us fix a partition $\Delta$ such that $\operatorname{diam}(\Delta)\leq\delta$ and consider the control $u(\cdot)=u(\cdot\mid t,w(\cdot),U^{\circ},\Delta)$ and the motion $x(\cdot)=x(\cdot\mid t,w(\cdot),T,u(\cdot))$ of system Eq. 7 formed by the control law $\{U^{\circ},\Delta\}$. As in the proof of Lemma 11.2, let us define the function $\omega(\cdot)$ by Eq. 65. Due to Eq. 69, we have $\omega(T)-\omega(\theta)\leq\varepsilon/2$. Therefore, according to Eq. 68, in order to complete the proof, it is sufficient to verify that $\omega(\theta)-\omega(t)\leq\varepsilon/2$. Hence, by Eq. 67, it remains to show that, for every $\tau\in[t,\theta)$,

[TABLE]

Let $\tau\in[t,\theta)$, and let $j\in\overline{1,k}$ be such that $\tau\in[\tau_{j},\tau_{j+1})$. By the choice of $\delta$,

[TABLE]

Since, from Eqs. 60, 62, and 51, it follows that $u(\tau)=u_{j}=U^{\circ}(\tau_{j},x_{\tau_{j}}(\cdot))$ and

[TABLE]

then, due to Eq. 50, we obtain

[TABLE]

*Thus, Eq. 70 holds, and the lemma is proved. *

Proof 11.6** (Proof of Theorem 11.1).**

*Let $(t,w(\cdot))\in G$ be fixed. If $t=T$, then Eq. 61 is a consequence of Eqs. 58 and 30. Let us suppose that $t<T$. Then, due to Lemma 11.4, in accordance with Eq. 28, we get $\rho(t,w(\cdot))\leq\varphi(t,w(\cdot))$. Hence, applying Lemma 11.2, we conclude Eq. 61. Finally, the optimality of the strategy $U^{\circ}$ follows directly from Eqs. 61 and 11.4. The theorem is proved. *

From Theorems 8.5, 10.1, 11.1, and 30, we derive the following.

Corollary 11.7**.**

*If the value functional $\rho$ (see Eq. 28) is $ci$-smooth of the order $\alpha$, then the control strategy defined by $\rho$ according to Eq. 62 is optimal. *

Thus, this corollary gives a way of constructing an optimal control strategy in the case when the value functional $\rho$ satisfies the additional smoothness assumptions. Let us note that, in the general case, the desired control strategy can be constructed on the basis of the methods developed in [13, 15].

12 Example: complete solution

Let us illustrate the results obtained in the paper by solving the example considered in Section 4. Namely, let us calculate the value functional $\rho$ and find an optimal control strategy $U^{\circ}$ in the optimal control problem for the dynamical system

[TABLE]

and the cost functional

[TABLE]

First, let us consider the auxiliary functional

[TABLE]

Let us note that, in accordance with Eq. 25, $\rho_{\ast}$ is the value functional of the degenerate optimal control problem for the system $(^{C}D^{\alpha}x)(\tau)=0$, $x(\tau)\in\mathbb{R}$, $\tau\in[0,T]$, and the cost functional $J(t,w(\cdot),u(\cdot))=x(T\mid t,w(\cdot),T,u(\cdot))$. Therefore, $\rho_{\ast}$ is continuous by Theorem 8.5. Further, let us fix $(t,w(\cdot))\in G^{0}$ and show that $\rho_{\ast}$ is $ci$-differentiable of the order $\alpha$ at $(t,w(\cdot))$. For every $x(\cdot)\in X(t,w(\cdot))$ and $\tau\in(t,T)$, we have

[TABLE]

Let us denote

[TABLE]

Then, applying the integration by parts formula, we obtain

[TABLE]

Hence, taking into account that

[TABLE]

and

[TABLE]

we derive

[TABLE]

Here and below, by the same symbol $o(\tau-t)$, we denote different functions with the property that $o(\tau-t)\rightarrow 0$ when $\tau\downarrow t$. Thus, the functional $\rho_{\ast}$ is $ci$-differentiable of the order $\alpha$ at $(t,w(\cdot))$, and

[TABLE]

Moreover, due to Proposition 8.3, the functional $G^{0}\ni(t,w(\cdot))\rightarrow(T-t)^{\alpha-1}\in\mathbb{R}$ is continuous, and, therefore, $\rho_{\ast}$ is $ci$-smooth of the order $\alpha$.

Now, let us consider the functional

[TABLE]

where $(t,w(\cdot))\in G$. Let us show that $\varphi$ satisfies all the assumptions of Theorem 11.1. First of all, we note that continuity of $\varphi$ follows from continuity of $\rho_{\ast}$ and the functional $G\ni(t,w(\cdot))\rightarrow(T-t)^{\alpha}\in\mathbb{R}$ (see Proposition 8.3). Further, let us show that $\varphi$ is $ci$-differentiable of the order $\alpha$ at every $(t,w(\cdot))\in G^{0}$. Let us fix $(t,w(\cdot))\in G^{0}$ and $x(\cdot)\in X(t,w(\cdot))$ and consider the following cases.

Let us suppose that $\rho_{\ast}(t,w(\cdot))<-(T-t)^{\alpha}$. Then, due to continuity of the function $[0,T]\ni\tau\mapsto(\tau,x_{\tau}(\cdot))\in G$ (see Proposition 8.3), there exists $\delta\in(0,T-t)$ such that $\rho_{\ast}(\tau,x_{\tau}(\cdot))<-(T-\tau)^{\alpha}$ for $\tau\in(t,t+\delta]$. Hence, for $\tau\in(t,t+\delta]$, we derive

[TABLE]

and, consequently, taking Eq. 72 into account, we get

[TABLE]

Hence, $\varphi$ is $ci$-differentiable of the order $\alpha$ at $(t,w(\cdot))$, and

[TABLE]

Further, if $|\rho_{\ast}(t,w(\cdot))|<(T-t)^{\alpha}$, then it is clear that $\varphi$ is $ci$-differentiable of the order $\alpha$ at $(t,w(\cdot))$, and $\partial_{t}^{\alpha}\varphi(t,w(\cdot))=\nabla^{\alpha}\varphi(t,w(\cdot))=0$.

Now, let $\rho_{\ast}(t,w(\cdot))=-(T-t)^{\alpha}$. Then, there exists $\delta\in(0,T-t)$ such that, for every $\tau\in(t,t+\delta]$, only two cases are possible: either $|\rho_{\ast}(\tau,x_{\tau}(\cdot))|\leq(T-\tau)^{\alpha}$ or $\rho_{\ast}(\tau,x_{\tau}(\cdot))<-(T-\tau)^{\alpha}$. In the first case, we have $\varphi(\tau,x_{\tau}(\cdot))-\varphi(t,w(\cdot))=0$. In the second case, we derive

[TABLE]

So, $\varphi$ is $ci$-differentiable of the order $\alpha$ at $(t,w(\cdot))$, $\partial_{t}^{\alpha}\varphi(t,w(\cdot))=\nabla^{\alpha}\varphi(t,w(\cdot))=0$.

By similar arguments, we obtain that $\varphi$ is $ci$-differentiable of the order $\alpha$ at $(t,w(\cdot))$ such that $\rho_{\ast}(t,w(\cdot))\geq(T-t)^{\alpha}$, and

[TABLE]

Thus, the functional $\varphi$ is $ci$-differentiable of the order $\alpha$ at every $(t,w(\cdot))\in G^{0}$. Moreover, the analysis of the calculated above values of $\partial_{t}^{\alpha}\varphi(t,w(\cdot))$ and $\nabla^{\alpha}\varphi(t,w(\cdot))$ shows that the functionals $\partial_{t}^{\alpha}\varphi:G^{0}\rightarrow\mathbb{R}$ and $\nabla^{\alpha}\varphi:G^{0}\rightarrow\mathbb{R}$ are continuous, and, therefore, $\varphi$ is $ci$-smooth of the order $\alpha$.

In the considered optimal control problem, $\mathcal{H}(s)=-\Gamma(\alpha+1)|s|$, $s\in\mathbb{R}$ (see Eq. 51). Then, Hamilton–Jacobi–Bellman equation Eq. 50 takes the following form:

[TABLE]

By direct substitution of the values of $\partial_{t}^{\alpha}\varphi(t,w(\cdot))$ and $\nabla^{\alpha}\varphi(t,w(\cdot))$, we conclude that $\varphi$ satisfies this equation. Further, according to Eqs. 5, 71, and 73, we have

[TABLE]

and, therefore, $\varphi$ satisfies right-end condition Eq. 58. Consequently, by Theorem 11.1, $\varphi$ is the value functional of the considered optimal control problem, i.e., $\rho(t,w(\cdot))=\varphi(t,w(\cdot))$, $(t,w(\cdot))\in G$, and an optimal control strategy can be determined by

[TABLE]

where $(t,w(\cdot))\in G^{0}$. The example is completely solved.

Remark 12.1**.**

In the case $\alpha=1$, the obtained solution agrees with the classical one. Namely, since $\rho_{\ast}(t,w(\cdot))=\rho_{\ast}(t,w(t))=w(t)$, then

[TABLE]

where $(t,w(\cdot))\in G$, and

[TABLE]

*where $(t,w(\cdot))\in G^{0}$. *

13 Conclusions

In the paper, we have studied a Bolza-type optimal control problem for a fractional-order dynamical system. We have proposed to consider the value of this problem as a functional in a suitable space of histories of motions. We have proved that, within this approach, the dynamic programming principle is satisfied. Further, we have introduced a new notion of fractional coinvariant differentiation of functionals and associated the optimal control problem with a Hamilton–Jacobi–Bellman equation with the fractional coinvariant derivatives. Under certain smoothness assumptions, we have established a connection between the value functional and solutions to this equation. In particular, we have proposed a way of constructing an optimal control strategy. The obtained results have been illustrated by an example.

Nevertheless, it should be noted that the value functional of the considered optimal control problem may not possess the specified smoothness properties. Therefore, future research will be devoted to developing the theory of generalized (minimax and viscosity) solutions to the obtained Hamilton–Jacobi–Bellman equation and studying their connection with the value functional in the general non-smooth case.