Regular path-constrained time-optimal control problems in three-dimensional flow fields

TL;DR

This paper develops a computational approach for solving three-dimensional time-optimal control problems with state constraints in steady flow fields, using an indirect maximum principle method that relies on regularity conditions.

Contribution

It introduces a regularity-based numerical method for solving complex flow field control problems, ensuring continuity of multipliers and stable extremal computations.

Findings

Method successfully computes extremals in various flow fields.

Regularity condition guarantees multiplier continuity and numerical stability.

Approach applicable to diverse three-dimensional flow control scenarios.

Abstract

This article concerns a class of time-optimal state constrained control problems with dynamics defined by an ordinary differential equation involving a three-dimensional steady flow vector field. The problem is solved via an indirect method based on the maximum principle in Gamkrelidze's form. The proposed computational method essentially uses a certain regularity condition imposed on the data of the problem. The property of regularity guarantees the continuity of the measure multiplier associated with the state constraint, and ensures the appropriate behavior of the corresponding numerical procedure which, in general, consists in computing the entire field of extremals for the problem in question. Several examples of vector fields are considered to illustrate the computational approach.

Equations

$$T$$

$$\dot{x}=u+v(x),$$

$$x(0)=A,x(T)=B,$$

$$g(x)\le 0\text{for all }x,$$

$$h(u):=u_1^2+u_2^2+u_3^2-1\le 0.$$

$$\Gamma (x,u)=⟨\nabla g(x),u+v(x)⟩.$$

$$\bar{H}(x,u,\psi ,\mu )=⟨\psi ,u+v(x)⟩-\mu \Gamma (x,u),$$

$$\dot{\psi }(t)$$

$$u^{\ast }(t)$$

$$\underset{u\in U}{\max }\{\bar{H}(x^{\ast }(t),u,\psi (t),\mu (t))\}=\lambda \text{for all }t\in [0,T^{\ast }];$$

$$|\psi (t)-\mu (t)\nabla g(x^{\ast }(t))|>0\text{ for all }t\in [0,T^{\ast }].$$

$$|⟨\nabla g(x),v(x)⟩|<r\forall x:g(x)=0.$$

$$⟨\nabla g(x),u⟩=-⟨\nabla g(x),v(x)⟩.$$

$$g(x):=x_1^2+x_2^2-1.$$

$${\dot{\psi }}_1(t)$$

$$-{\psi }_3(t)\text{partialderivative}{v}_3^{\ast }{x}_1(t)+2\mu (t)u_1^{\ast }(t)+2\mu (t)v_1^{\ast }(t),$$

$${\dot{\psi }}_2(t)$$

$$-{\psi }_3(t)\text{partialderivative}{v}_3^{\ast }{x}_2(t)+2\mu (t)u_2^{\ast }(t)+2\mu (t)v_2^{\ast }(t),$$

$${\dot{\psi }}_3(t)$$

$$-{\psi }_3(t)\text{partialderivative}{v}_3^{\ast }{x}_3(t).$$

$$u_1^{\ast }=\frac{{\psi }_1-2\mu x_1^{\ast }}{\sqrt{({\psi }_1-2\mu x_1^{\ast }{)}^2+({\psi }_2-2\mu x_2^{\ast }{)}^2+{\psi }_3^2}}$$

$$u_2^{\ast }=\frac{{\psi }_2-2\mu x_2^{\ast }}{\sqrt{({\psi }_1-2\mu x_1^{\ast }{)}^2+({\psi }_2-2\mu x_2^{\ast }{)}^2+{\psi }_3^2}}$$

$$u_3^{\ast }=\frac{{\psi }_3}{\sqrt{({\psi }_1-2\mu x_1^{\ast }{)}^2+({\psi }_2-2\mu x_2^{\ast }{)}^2+{\psi }_3^2}}.$$

$$(x_1^{\ast }(t){)}^2+(x_2^{\ast }(t){)}^2=1,\Gamma (x^{\ast }(t),u^{\ast }(t))=0,$$

$$\Gamma (x,u)=2x_1(u_1+v_1(x))+2x_2(u_2+v_2(x)).$$

$$\begin{array}[]{l}\mu^{2}-\mu(x_{1}^{*}\psi_{1}+x_{2}^{*}\psi_{2})\displaystyle+\,\frac{(x_{1}^{*}\psi_{1}+x_{2}^{*}\psi_{2})^{2}-|\psi|^{2}(x_{1}^{*}v_{1}^{*}+x_{2}^{*}v_{2}^{*})^{2}}{4(1-(x_{1}^{*}v_{1}^{*}+x_{2}^{*}v_{2}^{*})^{2})}=0.\end{array}$$

$$\begin{array}[]{l}\displaystyle\mu=\frac{1}{2}(x_{1}^{*}\psi_{1}+x_{2}^{*}\psi_{2})\pm\,\frac{1}{2}|x_{1}^{*}v^{*}_{1}+x_{2}^{*}v_{2}^{*}|\sqrt{\frac{|\psi|^{2}-(x_{1}^{*}\psi_{1}+x_{2}^{*}\psi_{2})^{2}}{1-(x_{1}^{*}v^{*}_{1}+x_{2}^{*}v_{2}^{*})^{2}}}.\end{array}$$

$$\begin{array}[]{l}\displaystyle\mu=\frac{1}{2}(x_{1}^{*}\psi_{1}+x_{2}^{*}\psi_{2})+\,\frac{1}{2}|x_{1}^{*}v^{*}_{1}+x_{2}^{*}v_{2}^{*}|\sqrt{\frac{|\psi|^{2}-(x_{1}^{*}\psi_{1}+x_{2}^{*}\psi_{2})^{2}}{1-(x_{1}^{*}v^{*}_{1}+x_{2}^{*}v_{2}^{*})^{2}}}.\end{array}$$

$$g(x):=x_1^2+x_2^2+x_3^2-1.$$

$${\dot{\psi }}_1(t)$$

$$+\left(-{\psi }_3(t)+2\mu (t)x_3^{\ast }(t)\right)\text{partialderivative}{v}_3^{\ast }{x}_1(t)+2\mu (t)u_1^{\ast }(t)+2\mu (t)v_1^{\ast }(t),$$

$${\dot{\psi }}_2(t)$$

$$+\left(-{\psi }_3(t)+2\mu (t)x_3^{\ast }(t)\right)\text{partialderivative}{v}_3^{\ast }{x}_2(t)+2\mu (t)u_2^{\ast }(t)+2\mu (t)v_2^{\ast }(t),$$

$${\dot{\psi }}_3(t)$$

$$+\left(-{\psi }_3(t)+2\mu (t)x_3^{\ast }(t)\right)\text{partialderivative}{v}_3^{\ast }{x}_3(t)+2\mu (t)u_3^{\ast }(t)+2\mu (t)v_3^{\ast }(t).$$

Full text

††journal: the European Journal of Control

url]www.nathaliekhalil.com

1 Introduction

This article concerns the study of time-optimal state-constrained problems and represents an extended version of [1]. In this article, a particular class of state-constrained time-optimal control problems is considered which satisfies a certain condition of regularity with respect to state constraints. Necessary optimality conditions in the form of the maximum principle and numerical techniques are brought together to shape an indirect method to solve the problem. The fundamental challenge encountered while applying indirect methods in the presence of state constraints is due to the fact that the Lagrange multiplier is a Borel measure whose support is embedded in the set of points in time at which the state trajectory meets the boundary of the state constraint. The singular component of this measure, more precisely, the atoms, prevent the correct execution of the proposed numerical algorithm. For this reason, in this work, we limit our study to the class of regular problems which satisfy the regularity assumptions with respect to the state constraints proposed in Chapter 6 of the classic monograph [2]. This property entails the absence of the atoms, – that is, the continuity of the measure-multiplier which has been examined, for example, in [3, 4, 5, 6, 7]. More precisely, a class of three-dimensional regular state-constrained time-optimal control problems affine with respect to the control and subject to a steady flow field is considered. A similar problem was investigated in [1] for the case of cylindrical state constraints. Our paper extends the results of [1] to the case of state constraint in the form of the unit sphere, and of a torus. Although the considered problems are cast in an abstract context, it is not difficult to imagine applications in which they might arise. Precision of the navigation of autonomous vehicles is a key feature for the success of its operation. The shape of the region in which the position error is below a pre-determined threshold defined in the system requirements depends on the localization method and this might be a cylinder, a sphere, a torus, or even some other more complex shape. Another important case consists in the data sampling underwater currents flowing in the oceanic water column with autonomous underwater vehicles. These underwater currents are, literally, cylinders of water with characteristics quite different from the surrounding water and the motion of the autonomous vehicle should be constrained to this cylinder.

Moreover, our paper extends the analysis in [8] where the two-dimensional case is exploited, to the three-dimensional framework. For the proposed optimal control problem, we apply the Pontryagin maximum principle and derive the associated two-point boundary value problem given with respect to the optimal control and the measure multiplier. The formulae for the optimal control and for the measure multiplier are derived from the maximum condition, while the expression for the measure multiplier essentially relies on the regularity. The optimal control and the measure multiplier are expressed in terms of the state and the adjoint variables. The two-point boundary value problem, which now depends only on the state variable and on the adjoint variable, being solved, leads to the determination of the extremals. The continuity of the measure multiplier is the key property enabling the computation of the junction points of these extremals.

The three-dimensional case considered here is more complex from computational point of view in comparison to the two-dimensional case studied in [8]. Solution to the corresponding boundary value problem requires to find two (instead of one in the two-dimensional case) parameters determining the initial value of the adjoint variable.

Optimal control problems in the presence of state constraints have been widely investigated in the literature. The classical theory in the field can be found, for example, in [2, 9, 10] while more recent studies complemented it with such important issues as non-degeneracy and normality of the maximum principle, non-smooth aspects, non-regular situations, etc., see, for example, [11, 12, 13, 14, 15, 16, 17], among others. Computational techniques to solve such problems and to find the set of extremals have also been broadly studied in the both framework of direct and indirect methods. Here we refer the reader to the sources [18, 19, 20, 21, 22, 23, 24, 25] for various issues on numerical solutions to state-constrained problems including direct and indirect approaches, and to [19, 21, 22, 23, 24, 25] where merely indirect numerical methods have been investigated. Numerical methods, closely related to the shooting method, to solve state constrained optimal control problems for nonlinear ODE, were investigated for instance in [26, 27]. In [26], first-order inequality state constraint are studied. Under some assumptions (for instance strong Legendre-Clebsch condition on the second order derivative of an augmented Hamiltonian, controllability, etc.), which are stronger than ours, the original problems with inequality constraints are locally replaced by some auxiliary problems with equality constraints. The classical implicit function theorem is applied to the latter problem, and sensitivity results are obtained. The measure multiplier of the original problem is constructed by using the Lagrange multipliers of the auxiliary problem, and in such a way it is continuous. In [27], the analysis is extended to consider second-order state constraint inequalities.

The article is organized as follows. In Section 2, the problem is formulated and the regularity concept is presented. In Section 3, the nondegenerate maximum principle for the investigated problem is stated. Section 4 concerns the application of the maximum principle for the cases in which the state constraint sets are cylinder, unit sphere, and torus. Explicit formulae for the extremal control and the measure multiplier are obtained for each case. In section 5, we describe the numerical method and illustrate it by showing computational results for the cylindrical and spherical state constraint cases. Finally, in Section 6, a brief conclusion and some perspectives of future research are given.

2 Problem formulation

We study a vehicle moving in a three-dimensional bounded and closed state domain defined by the given state constraints. The motion of the vehicle is affected by the presence of the fluid flow vector field $v(x)$, which intervenes in the dynamical control system. The path-constrained time-optimal control problem investigated in this article is as follows:

[TABLE]

Here, $x=(x_{1},x_{2},x_{3})$ is the state variable, $u=(u_{1},u_{2},u_{3})$ is the control variable. A measurable function $u(\cdot):[0,T]\to\mathbb{R}^{3}$ is termed control. The point $A$ is the starting point, while $B$ is the terminal point, and $v:\mathbb{R}^{3}\to\mathbb{R}^{3}$ is a smooth map defining a fluid flow varying in space. The terminal time $T$ is supposed to be minimized.

The state constraint is defined by the given function $g:\mathbb{R}^{3}\to\mathbb{R}$. Regarding this function, it is assumed in what follows that $\nabla g(x)\neq 0$ for all $x$ such that $g(x)=0$. Thus, the level set $g(x)=0$ is the so-called regular surface (manifold). Three different cases of the regular surface will be considered in the computational part of this work: cylinder, sphere and torus.

Consider the scalar product of the dynamics and the gradient of the state constraint function:

[TABLE]

If a feasible control process is considered in the arguments of $\Gamma$, then this function yields the total time derivative of the state constraint function with respect to the control differential system in study. Following [2, 6, 7], consider the a priori regularity condition imposed on the data of problem.

Regularity condition. Assume that for all $x\in\mathbb{R}^{3}$ and $u\in\mathbb{R}^{3}$, such that $g(x)=0$, $\Gamma(x,u)=0$, $h(u)=0$, the set of vectors $\partialderivative{\Gamma}{u}$ and $\nabla{h(u)}$ is linearly independent.

Remark 2.1**.**

It is simple to show that the regularity condition implies that the controllability conditions w.r.t. the state constraints, also known as IPC (Inward Pointing Condition) and OPC (Output Pointing Condition), are valid along any feasible trajectory (cf. [14] and the bibliography cited therein). These conditions intervene in proving the non-degeneracy of the maximum principle.

The next proposition provides conditions guaranteeing the existence of solution to Problem (LABEL:problem). The existence of solution is important for the forthcoming numerical analysis.

Proposition 2.2**.**

Assume that the regularity condition holds and that the vector field verifies $|v(x)|<1$ for all $x=(x_{1},x_{2},x_{3})$ such that $g(x)<0$. Assume also that $A$, and $B$ belong to the same connected component of the feasible state domain.

Then, Problem (LABEL:problem) has a solution.

Proof.

Problem (LABEL:problem) is linear while the velocity set is convex and compact. Therefore, in order to ensure the existence of a solution, due to Filippov’s theorem, [28], it is sufficient to justify the existence of at least a single feasible path connecting $A$ and $B$. Consider the connected component of the feasible state domain in which points $A$ and $B$ lie. If points $A$, and $B$ are in the interior of the feasible state domain, then such a path is guaranteed, as $A$ and $B$ can be connected by a smooth curve lying in the interior of the state constraint set, while the full controllability at each point of time is entailed by the condition $|v(x)|<1$. If one of the point $A$, or $B$, belongs to the boundary, then, by using the regularity of the flow field $v(x)$, which implies the controllability conditions (see Remark 2.1 above), there exists a feasible path from $A$ (or $B$) to some close to it point in the interior of the feasible state domain. Then, this case is reduced to the one already considered as the condition $|v(x)|<1$ once again intervenes at that “close” point to guarantee the existence of a feasible path from the starting point $A$ to the terminal point $B$. ∎

3 Maximum principle

For the maximum principle in Gamkrelidze’s form which we consider, the extended Hamilton-Pontryagin function is defined as:

[TABLE]

where $\psi\in\mathbb{R}^{3}$, $\mu\in\mathbb{R}$ are the adjoint variables.

Assume that the regularity condition holds. Then, for an optimal process $(x^{*},u^{*},T^{*})$, the maximum principle derived in [6] (see Theorem 4.5 therein) ensures the existence of Lagrange multipliers: a number $\lambda\in[0,1]$, an absolutely continuous adjoint arc $\psi=(\psi_{1},\psi_{2},\psi_{3})\in W_{1,\infty}([0,T^{*}];\mathbb{R}^{3})$, and a scalar continuous function $\mu\in C([0,T^{*}];\mathbb{R})$, such that the following conditions are satisfied:

• (a)

Adjoint equation

[TABLE]

• (b)

Maximum condition

[TABLE]

where $U:=\{u\in\mathbb{R}^{m}\ :\ h(u)\leq 0\}$;

• (c)

Conservation law

[TABLE]

where $U:=\{u\in\mathbb{R}^{m}\ :\ h(u)\leq 0\}$;

• (d)

$\mu(t)$ is decreasing, and constant on the time intervals where $g(x^{*}(t))<0$;

• (e)

Non-triviality condition

[TABLE]

Above, $T^{*}$ stands for the optimal time, thus, the optimal pair $(x^{*},u^{*})$ is considered over the time interval $[0,T^{*}]$.

4 Applications

In this part of work, we consider three particular cases of state constraint sets in $\mathbb{R}^{3}$. For each case, we derive the corresponding adjoint system, and explicit the expressions of the measure multiplier and the optimal control with respect to the state and adjoint variables.

The following simple assertion ensuring the regularity condition under some assumptions on $g$ and $v(x)$ facilitates the analysis of the on-going applications.

Proposition 4.1**.**

Assume that there exists a number $r\in\mathbb{R}\neq 0$ such that $|\nabla g(x)|=r$ for all $x:$ $g(x)=0$. Then, the regularity condition is satisfied whenever the following condition on the steady flow field $v(x)$ is imposed:

[TABLE]

Proof.

It is not restrictive to consider the case $r=1$. Consider any $x,u$ such that $g(x)=h(u)=\Gamma(x,u)=0$. Then,

[TABLE]

By assumption 2, it holds that $|\langle\nabla g(x),u\rangle|<1$. However, both vectors $\frac{1}{2}\nabla h(u)=u$ and $\nabla g(x)$ belong to the unit sphere. Therefore, the set of vectors $u$ and $\partialderivative{\Gamma}{u}(x,u)=\nabla g(x)$ is linearly independent.

∎

We study three cases of the state constraint set: cylinder, unit sphere, torus. The cylinder case was previously investigated in [1] and a sample problem was considered for a specific vector field.

4.1 Cylinder

Consider the case when the function $g$ representing the state constraint takes the following form

[TABLE]

Observe that by virtue of Proposition 4.1 the regularity condition is satisfied if the vector field verifies (2), that is, if the vector field verifies the estimate $|\langle x,v(x)\rangle|<1$ for all $x$ such that $g(x)=0$.

Next, we explicit the necessary conditions for a given optimal process $(x^{*},u^{*},T^{*})$ to Problem (LABEL:problem). The adjoint system implies (we use the notation $v^{*}(t)$ for $v(x^{*}(t))$, and also for its partial derivatives),

[TABLE]

[TABLE]

[TABLE]

Moreover, the maximum condition and the non-triviality condition allow us to uniquely find the optimal control $(u_{1}^{*},u_{2}^{*},u_{3}^{*})$ (here, the time dependence, for simplicity, is omitted):

[TABLE]

[TABLE]

[TABLE]

At the boundary of the state constraint, one has

[TABLE]

where

[TABLE]

Using these relations, replacing $u_{1}^{*},u_{2}^{*}$ and $u_{3}^{*}$ by their expressions in $\Gamma(x^{*},u^{*})=0$, we obtain the quadratic equation with respect to $\mu$:

[TABLE]

The solutions are:

[TABLE]

However, the acceptable solution is:

[TABLE]

The derived expression for $\mu$ holds at the boundary of the state constraints, that is, on the time intervals on which $g(x^{*}(t))=0$. Moreover, function $\mu(t)$ is continuous, decreasing, and it is constant on the intervals where $g(x^{*}(t))<0$.

4.2 Unit sphere

For the unit sphere case, the function $g$ is:

[TABLE]

Observe that by virtue of Proposition 4.1 the regularity condition is satisfied if the vector field verifies (2), that is, if the vector field verifies the estimate $|\langle x,v(x)\rangle|<1$ for all $x$ such that $g(x)=0$.

The adjoint system in this case gives:

[TABLE]

[TABLE]

[TABLE]

The maximum condition permits to uniquely define the optimal controls:

[TABLE]

[TABLE]

[TABLE]

At the boundary of the state constraint, one has

[TABLE]

where

[TABLE]

By following the same reasoning as in the case of the cylinder, and replacing $u_{1}^{*},u_{2}^{*}$ and $u_{3}^{*}$ by their expressions in $\Gamma(x^{*},u^{*})=0$, we obtain the following expression of $\mu$:

[TABLE]

The acceptable solution is:

[TABLE]

The derived expression for $\mu$ holds at the boundary of the state constraints, that is, on the time intervals on which $g(x^{*}(t))=0$. Moreover, function $\mu(t)$ is continuous, decreasing, and it is constant on the intervals where $g(x^{*}(t))<0$.

4.3 Torus

A more non-trivial state constraint is represented by the torus symmetric about the $x_{3}$-axis. In this case, the function $g$ is:

[TABLE]

where $R$ is the so-called major radius.

Observe that by virtue of Proposition 4.1 the regularity condition is satisfied if the vector field verifies (2), that is if, $|wx_{1}v_{1}+wx_{2}v_{2}+x_{3}v_{3}|<1$ for all $x$ such that $g(x)=0$, where $w:=\frac{\sqrt{x_{1}^{2}+x_{2}^{2}}-R}{\sqrt{x_{1}^{2}+x_{2}^{2}}}.$

Denote by $w^{*}:=\frac{\sqrt{(x_{1}^{*})^{2}+(x_{2}^{*})^{2}}-R}{\sqrt{(x_{1}^{*})^{2}+(x_{2}^{*})^{2}}}$. The adjoint system in this case gives:

[TABLE]

[TABLE]

[TABLE]

From the maximum condition, we obtain the optimal controls:

[TABLE]

[TABLE]

[TABLE]

We have

[TABLE]

where $\displaystyle w:=\frac{\sqrt{x_{1}^{2}+x_{2}^{2}}-R}{\sqrt{x_{1}^{2}+x_{2}^{2}}}.$ Therefore, at the boundary of the state constraint, the following quadratic equation with respect to $\mu$ arises:

[TABLE]

The acceptable solution is

[TABLE]

The derived expression for $\mu$ holds at the boundary of the torus. Moreover, function $\mu(t)$ is continuous, decreasing, and it is constant on the intervals where $x^{*}(t)$ is within the torus interior. Note that when $R=0$, we recover the formulae in the case of the unit sphere.

5 Numerical results

The proposed computational algorithm is based on the maximum principle, that is, on the conditions (a)–(e) stated in section 3. As it has been shown in the previous section, the regularity condition enabled us to obtain the expression for the measure multiplier through the state and adjoint variables $x(t)$ and $\psi(t)$ respectively. Upon substitution of this expression as well as the ones for extremal controls into the adjoint system, a two-point boundary-value problem is obtained:

[TABLE]

together with (4.1)-(9) for the cylinder or (4.2)-(16) for the unit sphere case. Here, the fluid flow $v(x)$, the starting point, $A$, and the terminal point, $B$, are given, and $T^{*}$, the optimal travelling time, is unknown. The problem is solved numerically by a variant of the shooting method described below. See for instance [29, 30] for an overview on the numerical methods for two-points boundary-value problems.

The measure multiplier $\mu(t)$ is non-constant only when the corresponding trajectory $x(t)$ lies at the boundary of the state constraint. The continuity of the measure multiplier is used for computation of the junction points, that is the points where the extremal arc meets the boundary of the state constraint.

Let us briefly outline the numerical algorithm used for the computation of the field of extremals.

We set $\mu(0)=0$. In this regard, see Remark 3.1 in [31]. Then, condition (e) featured in section 3 enables us to consider the initial value for $\psi$ from the surface of the unit sphere, $|\psi(0)|=1$. This unit sphere surface is parameterized by the two angles $\theta\in[0,\pi]$ and $\phi\in[0,2\pi)$:

[TABLE]

For a given value of $\theta$ and $\phi$, the system of governing equations is integrated numerically by the fourth-order Runge-Kutta method. Using the bisection method in $\theta$, and $\phi$, the trajectories not meeting the boundary and satisfying $|x(T^{*})-B|<10^{-3}$ are computed, being $10^{-3}$ the required accuracy.

The bisection method is also used to compute the junction points of the trajectories meeting the boundary, but only those trajectories for which $\mu$ is continuous at the junction point, i.e. $|\mu|<10^{-3}$, are selected. For such trajectories, the equations governing the overall system are integrated further in time, and, according to (6)–(9) for the cylinder and to (13)–(16) for the unit sphere case, the resulting trajectory follows the boundary and never leave it. Integrating the system along the boundary, at each time step, $\bar{t}$, we also compute trajectories “leaving” the boundary – corresponding to the initial conditions $x=x(\bar{t}),$ $\psi=\psi(\bar{t})$ and assuming $\mu=\mu(\bar{t})$ to be constant for all $t>\bar{t}$. If such trajectory satisfies $|x(T^{*})-B|<10^{-3}$ for $T^{*}>\bar{t}$, it belongs to an extremal, together with the corresponding boundary segment and the trajectory entering the boundary.

In the first example, we apply this numerical procedure to the Problem (LABEL:problem) with the state constraints given by the cylinder (see section 4.1) for the fluid flow $v(x)=(0,\,0,\,x_{1}^{2}+x_{2}^{2})$ satisfying the regularity condition (2), $A=(0.2,\,-0.5,\,0)$ and $B=(0,\,0.5,\,5)$. This vector field $v(x)$, which corresponds to a fluid flowing faster on the boundary of the cylinder, is chosen to obtain extremals with active boundary. Indeed, travelling along the boundary is more beneficial, in the sense that such trajectories take less time to join their endpoints. The set of extremals is shown in the space of $(x_{1},\,x_{2},\,x_{3})$ in Figure 1, as well as its projection onto the plane $x_{3}=0$. The set is constituted of two extremals – one containing a boundary segment (red line), and one not meeting the boundary (black line).

The optimal extremal (red line) has a boundary segment (for $0.92\leq t\leq 1.78$), the corresponding travelling time is $T^{*}=3.81$; the path from $A$ to $B$ along the extremal not meeting the boundary (black line) takes $4.25$ time units. Time evolution of each component of both extremals is shown in Figure 2. For the optimal extremal, the evolution of the control, $u^{*}(t)$, as well as of the Lagrange multiplier $\mu(t)$ is also shown in Figure  2.

By construction, the multiplier $\mu(t)$ is constant when the trajectory is in the interior of the state constraint (i.e. for $t<0.92$ and $t>1.78$). Along the boundary (see right panel of Figure 2), the measure multiplier demonstrates linear behavior with respect to time. This can be shown analytically for the considered fluid flow.

Next, we consider the spherical case described in section 4.2. We take the following flow representing a horizontal vortex,

[TABLE]

satisfying the regularity condition (2). The starting and terminal points are chosen to be $A=(0.6,\,0.6,\,0.4)$ and $B=(-0.6,\,-0.6,\,0)$, respectively. The set of computed extremals, shown in Figure 3, is represented by three curves, two of them do not meet the boundary (shown in black and red) and one meeting the boundary (shown in blue). Evolution of coordinates of the extremals not meeting the boundary as well as controls for the optimal extremal are shown in Figure 4.

Travelling along the extremal shown in black takes 1.73, while travelling along the optimal extremal (shown in red) is roughly twice faster (0.81 time units). Although the flow is faster closer to the boundary, travelling along it is not favorable (as in the previous example), it takes 1.98 time units.

6 Conclusions

In this paper, a time-optimal control problem in a three-dimensional steady vector flow field is analyzed in the presence of a state constraint given in the form of cylinder, unit sphere, and torus. Regularity conditions with respect to the state constraints and the vector flow field in study are considered. The maximum principle is applied to the problem in question under these regularity conditions. It is shown how the regularity condition, which implies the continuity of the measure multiplier, assists in solving numerically the boundary-value problem associated to the maximum principle. In this context, explicit formulae for the measure multiplier and the extremal controls are obtained as expressions with respect to the state and adjoint functions. These expressions are further substituted into the boundary-value problem, and solved by a variant of the shooting method. The obtained results are numerically illustrated for several three-dimensional sample vector flow fields and state constraints, and the corresponding set of extremals is plotted.

There are several avenues to extend this work. One possible way may consist in decreasing the degree of smoothness of the mapping defining the state constraints. Note that the proposed computational method essentially relies on the existence of the second order derivative of the mapping $g$. At the same time, an appropriate approximation sequence of smooth problems may assist in solving numerically the control problem in which $g$ is of merely $C^{1}$-class. Another avenue may consist in considering vector-valued state constraints. There are a number of possibilities to address this problem that the research effort should consider. Still, another direction is the consideration of general nonlinear dynamics. In this case, we might study problems for which the current framework holds and problems for which regularization techniques have to be applied. Finally, here the vector field mapping is defined a priori. There are interesting problems, for which this not the case, and the control action may affect the evolution of the vector field mapping. This latter problem is of significant complexity but also relevant for many applications.

Acknowledgements

The first two authors are supported by the Russian Science Foundation during the project 19-11-00258 carried out in the Federal Research Center “Informatics and Control” of the Russian Academy of Sciences. The valuable support of FCT (Portugal), research funding granted to the SYSTEC R&D Unit under project NORTE-01-0145-FEDER-000033 – STRIDE (COMPETE 2020), is also highly acknowledged.