Necessary conditions for turnpike property for generalized linear–quadratic problems

Abstract

In this paper, we develop several necessary conditions of turnpike property for generalized linear–quadratic (LQ) optimal control problem in infinite-dimensional setting. The term ‘generalized’ here means that both quadratic and linear terms are considered in the running cost. The turnpike property reflects the fact that over a sufficiently large time horizon, the optimal trajectories and optimal controls stay for most of the time close to a steady state of the system. We show that the turnpike property is strongly connected to certain system theoretical properties of the control system. We provide suitable conditions to characterize the turnpike property in terms of the detectability and stabilizability of the system. Subsequently, we show the equivalence between the exponential turnpike property for generalized LQ and LQ optimal control problems.

Appendix A: The closed-form solution of the optimal control

Lemma A.1

For any $T>0$ and $x_0\in \mathcal {H}$, if there exists a vector $w\in D(A^*)$ and a steady state $(x_e,u_e)$ such that

$$\begin{aligned} \begin{bmatrix} A^*\\ B^* \end{bmatrix} w= \begin{bmatrix} z+C^*Cx_e\\ v+K^*Ku_e \end{bmatrix} , \end{aligned}$$

(A.1)

then the optimal pair $(x^*_T(\cdot ,x_0),u^*_T(\cdot ,x_0))$ of problem $(GLQ)_T$ satisfies

$$\begin{aligned} \begin{aligned} u_T^*(t,x_0)-u_e=-(K^*&K)^{-1}B^*P(T-t)(x_T^*(t,x_0)-x_e)\\&-(K^*K)^{-1}B^*(U_{T-t}(T-t,0))^*w,\;\;\forall t\in [0,T]. \end{aligned} \end{aligned}$$

(A.2)

Proof

For any $T>0$, we claim that $(U_T(T,0))^*$ is the evolution operator at time T of the following evolution system:

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\textrm{d}p(t)}{\textrm{d}t}=(A^*-P(t)B(K^*K)^{-1}B^*)p(t),\\&p(0)=p_0\in \mathcal {H}. \end{aligned} \right. \end{aligned}$$

In other words, we have that $(U_T(T,0))^*p_0=p(T)$.

The proof is based on Yosida approximations. Let $A_n:=nA(nI-A)^{-1}\in \mathcal {L}(\mathcal {H})$ denote the Yosida approximation of A for sufficiently large $n\in \mathbb {N}$. Assume that $x_0,p_0\in \mathcal {H}$. For each n, we let $x_n$ denote the solution of problem

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\textrm{d}x_n(t)}{\textrm{d}t}=(A_n-B(K^*K)^{-1}B^*P(T-t))x_n(t),\;\;t\in [0,T]\\&x_n(0)=x_0, \end{aligned} \right. \end{aligned}$$

and $p_n$ denote the solution of problem

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\textrm{d}p_n(t)}{\textrm{d}t}=(A_n^*-P(t)B(K^*K)^{-1}B^*)p_n(t),\;\;t\in [0,T]\\&p_n(0)=p_0. \end{aligned} \right. \end{aligned}$$

Since $A_n$ is bounded, we can easily verify (by showing that the derivative is 0) that

$$\begin{aligned} \langle x_0,p_n(T)\rangle =\langle x_n(T),p_0\rangle . \end{aligned}$$

By [3, Part II, Chapter 1, Proposition 3.4], $p_n(T)\rightarrow p(T)$ and $x_n(T)\rightarrow U_T(T,0)x_0$ as $n\rightarrow \infty$. This further implies that

$$\begin{aligned} \langle x_0,p(T)\rangle =\langle U_T(T,0)x_0,p_0\rangle =\langle x_0,(U_T(T,0))^*p_0\rangle . \end{aligned}$$

Since $x_0,p_0\in \mathcal {H}$ can be chosen arbitrarily, our claim then follows.

Now fix some $T>0$ and define $p(\cdot )=(U_{\cdot }(\cdot ,0))^*w$ on [0, T]. Since $w\in D(A^*)$, by [3, Part II, Chapter 1, Proposition 3.5] we have that

$$\begin{aligned} p(\cdot )\in C([0,T],D(A^*))\cap C^1([0,T],\mathcal {H}). \end{aligned}$$

We then deduce that

$$\begin{aligned} \frac{\textrm{d}p(T-t)}{\textrm{d}t}=-(A^*-P(T-t)B(K^*K)^{-1}B^*)p(T-t)\;\;\text {in}\;\;\mathcal {H},\;\;\forall t\in [0,T]. \end{aligned}$$

Let x be the solution of problem (2.3) corresponding to input $u\in L^2(0,T;\mathcal {U})$ and initial condition $x_0\in \mathcal {H}$. Then, thanks to assumption (A.1),

$$\begin{aligned} \int _0^T\langle z,x(t)&\rangle +\langle v,u(t)\rangle \textrm{d}t\\&=\int _0^T\langle w,Ax(t)+Bu(t)\rangle _{\mathcal {H}_1^d,\mathcal {H}_{-1}}-\langle Cx_e,Cx(t)\rangle -\langle Ku_e,Ku(t)\rangle \textrm{d}t\\&=\langle w,x(T)-x_0\rangle +\int _0^T-\langle Cx_e,Cx(t)\rangle -\langle Ku_e,Ku\rangle \textrm{d}t. \end{aligned}$$

Combining this with (3.3) shows that

$$\begin{aligned} \begin{aligned} J_T(x_0,u)&=\int _0^T\Vert K\{(u(t)-u_e)+(K^*K)^{-1}B^*P(T-t)(x(t)-x_e)\}\Vert ^2\textrm{d}t\\&\quad -J_T(x_e,u_e)+\langle P(T)(x_0-x_e),(x_0-x_e)\rangle +2{\text {Re}}\langle w,x(T)-x_0\rangle . \end{aligned} \end{aligned}$$

(A.3)

On the other hand, we have

$$\begin{aligned} 2{\text {Re}}\langle&x(T),w\rangle -2{\text {Re}}\left\langle x_0,p(T)\right\rangle \\&={\text {Re}}\int _0^T2\left\langle Ax(t)+Bu(t),p(T-t)\right\rangle _{H_{-1},H_{1}^d}\\&\quad +2\left\langle x(t),-(A^*-P(T-t)B(K^*K)^{-1}B^*)p(T-t)\right\rangle \textrm{d}t\\&={\text {Re}}\int _0^T2\left\langle u(t),B^*p(T-t)\right\rangle \\&\quad +2\left\langle (K^*K)^{-1}B^*P(T-t)x(t),B^*p(T-t)\right\rangle \textrm{d}t. \end{aligned}$$

This, together with (A.3) implies that

$$\begin{aligned} J_T(x_0,u)&=\int _0^T\Vert K\{(u(t)-u_e)+(K^*K)^{-1}B^*P(T-t)(x(t)-x_e)\}\Vert ^2\textrm{d}t\\&\quad +2{\text {Re}}\langle w,x(T)\rangle +M_0\\&=\int _0^{T} \Vert K\{(u(t)-u_e)+(K^*K)^{-1}B^*P(T-t)(x(t)-x_e)\}\Vert ^2\\&\quad +2{\text {Re}}\left\langle (K^*K)^{-1}B^*P(T-t)(x(t)-x_e),B^*p(T-t)\right\rangle \\&\quad +2{\text {Re}}\left\langle (u(t)-u_e),B^*p(T-t)\right\rangle dt+M_1\\&=\int _0^{T}\left\| K\left\{ (u(t)-u_e)+(K^*K)^{-1}B^*\left[ P(T-t)(x(t)-x_e)+p(T-t)\right] \right\} \right\| ^2\textrm{d}t\\&\quad +M_2 \end{aligned}$$

where $M_0$, $M_1$ and $M_2\in \mathbb {R}$ are constants independent of u.

This implies that, if the closed loop system corresponding to the feedback law

$$\begin{aligned} u(t)-u_e&=-(K^*K)^{-1}B^*P(T-t)(x(t)-x_e)\\&-(K^*K)^{-1}B^*(U_{T-t}(T-t,0))^*w,\;\;\forall t\in [0,T], \end{aligned}$$

admits a solution in $L^2(0,T;\mathcal {H})$, then this solution is optimal.

By [3, Part II, Chapter 1, Proposition 3.5], this problem does admit a unique solution x in $L^2(0,T;\mathcal {H})$. So, the optimal pair verifies equation (A.2). $\square$