LQ Optimal Control of First-Order Hyperbolic PDE Systems with Final State Constraints

Partial differential equations (PDEs) have extensive applications in the real world, which are classified into different types according to evolution natures, consisting of hyperbolic, parabolic, elliptic, etc. In particular, the first-order hyperbolic PDEs, as one of the most important types of PDEs, are commonly used to characterize various physical phenomena and social behaviors, such as traffic flow [1], fuel cell control [2], cardiovascular flow analysis [3]. Therefore, broad attention has been paid to the first-order hyperbolic PDEs and plenty of progress has been made [4, 5, 6].

Two fundamental problems have been stuided for the systems governed by first-order hyperbolic PDEs, one is the optimal control problem, and the other is the controllability problem. On one hand, for the optimal control problem of first-order hyperbolic PDE system, [7] and [8] studied the linear-quadratic (LQ) problem and designed the optimal internal control and optimal boundary control based on operator Riccati equations, respectively. In [9], the optimal control problem was studied by the Galerkin finite element method, which is a kind of numerical methods, and the corresponding posterior error estimates was obtained. Combining the advantages of the above two methods, [10] provided a computable explicit solution of the LQ optimal control problem. By reinforcement learning technique, an online policy iteration algorithm for optimal control of first-order hyperbolic PDE systems was designed in [11]. On the other hand, the controllability problem aims to explore whether a dynamical system could be driven by controllers toward the desired final state from an arbitrary initial state. In [12] and [13], the exact controllability problems of the first-order hyperbolic PDE systems were studied with the internal control and the boundary control, respectively. The stochastic case was studied in [14, 15] and the corresponding exact controllability results have been obtained. Based on Carleman estimates, the exact controllability of the time dependent hyperbolic systems was studied in [16]. The minimal time for the exact controllability of first-order hyperbolic systems was presented in [17, 18].

It is noticed that either the optimal control problem without final state constraints, or the controllability problem with the given final state conditions was studied in the above literature. For the optimal control problem of PDE systems with final state constraints, [19] studied the parabolic equation, and [20] considered the wave equation. However, to the author’s best knowledge, there has been no substantial progress on the first-order hyperbolic PDEs.

In this paper, the first-order hyperbolic PDE optimal control problem with final state constraints is studied, where the cost function is considered as quadratic form. The Lagrange multiplier method and the technique of solving forward and backward partial differential equations (FBPDEs) are applied to solve this problem. The main contributions of this paper are as follows. 1) The solvability condition for the LQ optimal control problem of first-order hyperbolic PDE systems with final state constraints is presented. 2) The corresponding optimal control is given explicitly in the form of state feedback, where a non-homogeneous relationship is established between the optimal costate and the optimal state. Moreover, the result is reduced to a special case of the zero-valued final state constraints.

The rest of the paper is organized as follows. The problem to be addressed is stated in Section II. The main results in this paper are presented in Section III. Several numerical examples are provided in Section IV. Finally, section V concludes the paper.

Notations. $\mathbb{R}^{n}$ $(n\geq 1)$ denote $n$-dimensional real Euclidean space, $\mathbb{R}=\mathbb{R}^{1}.$ $x_{t}(z,t)$ and $x_{z}(z,t)$ denote the partial derivatives of the variables $t$ and $z$, respectively. For a subset $X$ of $\mathbb{R}^{n}$ $(n\geq 1)$, the following function spaces are useful in the sequel of the paper:

where

Consider the following linear system governed by first-order hyperbolic PDE:

for $z\in[0,l],t\in[0,T]$, $l>0,T>0,$ where $x(z,t)\in\mathbb{R}$ is state, $u(z,t)\in\mathbb{R}$ is input control, $a,$ $b$ are constants, and $c>0$. The initial data $\varphi(z)\in C^{1}[0,l]$ and the boundary data $\phi(t)\in C^{1}[0,T]$ are given with $\varphi(l)=\phi(0)$ for the compatibility at the point $(z,t)=(l,0)$ (i.e., the intersection of the domains of definition). For any continuously differentiable $u(z,t)$, system (1) admits a unique $C^{1}$ solution, which follows from the PDE well-posedness theory in [21, 22].

The quadratic cost function is considered as

where constants $q\geq 0$, $r>0$, and $p(z)\in C^{1}[0,l]$ with $p(0)=0$ is a given nonnegative-valued function.

To present the final state constraints on system (1) reasonably well, we introduce the following Riccati equation

for $z\in[0,l],t\in[0,T]$, with $\bar{b}=-\frac{b^{2}}{r}$. Provided that (3) has a solution $g(z,t)$, we define the following functions:

for $z\in[0,l],t\in[0,T]$, and

for $z\in[0,l],t\in[\frac{l-z}{c},T]$.

The addressed LQ optimal control problem with final state constraints is stated as follows.

To tackle this type of constrained optimal control problem, the first step is to convert it into a problem without final state constraints by using the Lagrange multiplier method. Specifically, by introducing an augmented functional

for $T>\frac{2l}{c},$ where $\gamma(z)$ can be an arbitrary constant-valued function, then Problem 1 can be restated as below:

Given that Problem 1 has been converted into Problem 2, we first present the solvability result of Problem 2 as follows.

Proof. By defining $J(u)$ derived by arbitrary controller $u(z,t)$ with the associated state $x(z,t)$ satisfying (1), and defining $J(u^{*})$ derived by (II) with the controller $u^{*}(z,t)$ and the state $x^{*}(z,t)$ satisfying FBPDEs (10)-(14), now we show that

In view of cost function (II), it follows that

where the last inequality is due to the fact that the values of $q$, $r$, and $p(z)$ are nonnegative. By substituting (11), (13) and (14) into (III), and using integration by parts, we derive that

Combining with (1), (10), (12), and (13), it is obtained from (III) that

Thus, (15) holds. This implies that the controller $u^{*}(z,t)$ (14) is optimal, and thus Problem 2 is solvable. The proof is now completed. $\Box$

Observed that the solvability of Problem 2 with the arbitrary given $\gamma(z)$ has been provided in Lemma 1. To obtain the solvability of Problem 1, in what follows, we define the function

for $z\in[0,l],t\in[\frac{l-z}{c},T]$, and further introduce the equation

for $z\in[0,l],t\in[0,T]$. Then, the main results of Problem 1 are presented as follows.

Proof. See Appendix A for more details of the proof. $\Box$

In Theorem 1, the optimal control result under the final state data $\eta(z)$ has been provided. Now we reduce the result into the case of zero-valued final state constraints to show the wider application of Theorem 1. Specifically, for a class of commonly applied boundary conditions $\phi(t)=0$ (eg. [23]), system (1) is rewritten as

Then the corresponding result to system (24) and the explicit optimal controller are provided in the following corollary.

Proof. By letting $\phi(t)=0$ in system (1) and $\gamma(z)=0$ in (18), Corollary 1 follows from Theorem 1. $\Box$

In this section, several numerical examples are simulated to verify Theorem 1, where the simulations are performed on the time interval $[0,6]$ and the spatial interval $[0,1]$ through the finite difference method with the step size $\tau=0.001$ and $h=0.001.$

Firstly, system (1) and cost function (II) are considered with the following parameters

The initial data and boundary data are given by

for $z\in[0,1],t\in[0,6]$. The final conditions are considered as

By solving Riccati equation (26) and calculating (8), we obtain that $\eta(0)=2.2737$ in Problem 1.Then, we design the optimal controller by (20) and simulate the corresponding state trajectory by (1). As shown in Fig. 2, it can be observed that the state trajectory under the designed controller reaches the given final conditions (30) at the final time $t=6.$ For comparison, the corresponding state trajectory without controller is depicted in Fig. 2.

To illustrate the powerful applicability of the result in this paper for various types of data, we present another numerical example by reselecting the initial data and boundary data as

and the final condition as the following piecewise function

where $\eta(0)=0.4547$ can be obtained by (3) and (8). Then, by designing the optimal controller according to (20), the state trajectories with and without the designed controller are shown in Fig. 4 and Fig. 4, respectively. It can be seen that the final state satisfies (32) under the designed controller as shown in Fig. 4.

In this paper, the constrained LQ optimal control problem of first-order hyperbolic PDE systems has been studied. The optimal controllers and the corresponding optimal cost of this problem have been explicitly given. The result has been reduced to the case of zero-valued final state constraints. Finally, several numerical examples have been simulated to verify the main results.