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SCIENCE CHINA Information Sciences, Volume 62, Issue 9: 192203(2019) https://doi.org/10.1007/s11432-018-9685-8

Optimal control with irregular performance

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  • ReceivedAug 23, 2018
  • AcceptedSep 30, 2018
  • PublishedAug 2, 2019

Abstract

In this paper, we solve the long-standing fundamental problem of irregular linear-quadratic (LQ) optimal control, which has received significant attention since the 1960s.We derive the optimal controllers via the key technique of finding the analytical solutions to two different forward and backward differential equations (FBDEs).We give a complete solution to the finite-horizon irregular LQ control problem using a new `two-layer optimization' approach. We also obtain the necessary and sufficient condition for the existence of optimal and stabilizing solutionsin the infinite-horizon case in terms of solutions to two Riccati equationsand the stabilization of one specific system.For the first time, we explore the essential differences between irregular and standard LQ control, making a fundamental contribution to classical LQ control theory.We show that irregular LQ control is totally different from regular control as the irregular controller must guarantee the terminal state constraint of $P_1(T)x(T)=0$.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61633014, 61573221, 61873332) and the Qilu Youth Scholar Discipline Construction Funding from Shandong University.


Supplement

Appendix

Proof of Lemma 3.2

Proof of necessity. Based on the discussion of (9)–(11), we can see that $p(t)\neq~P(t)x(t)$ under the condition (11), where $P(t)$ is the solution to (8). We therefore define a new variable $\Theta(t)$ as \begin{eqnarray}p(t)=P(t)x(t)+\Theta(t), \tag{36} \end{eqnarray} where it is clear that $\Theta(T)=0$. Next, we aim to derive the new FBDEs (16)–(18) under the solvability of Problem 1.

First, we take the derivative of (36), obtaining \begin{eqnarray}\dot{p}(t)=\dot{P}(t)x(t)+P(t)\big[Ax(t) +Bu(t)\big]+\dot{\Theta}(t). \tag{37} \end{eqnarray} From (6) and (36), we then find that \begin{eqnarray}\dot{p}(t)=-\big[A'P(t)x(t)+A'\Theta(t) +Qx(t)\big]. \tag{38} \end{eqnarray} By comparing (37) and (38), we obtain \begin{eqnarray}0&=&\dot{P}(t)x(t)+P(t)Ax(t)+P(t)Bu(t)+\dot{\Theta}(t) +A'P(t)x(t)+A'\Theta(t)+Qx(t). \tag{39} \end{eqnarray} Second, we aim to find the controller $u(t)$ and the new equilibrium condition (16). By using (36), we can formulate the equilibrium condition (7) as \begin{eqnarray}0=Ru(t)+B'p(t)=Ru(t)+B'P(t)x(t)+B'\Theta(t). \tag{40} \end{eqnarray} Taken together with (11), this can also be written as \begin{eqnarray}u(t)&=&-R^{\dag}\left(B'P(t)x(t)+B'\Theta(t)\right)+(I-R^{\dag}R)z(t), \tag{41} \end{eqnarray} where $z(t)$ is an arbitrary vector with compatible dimension such that the following equality holds: \begin{eqnarray}0&=&(I-RR^{\dag})\left(B'P(t)x(t)+B'\Theta(t)\right). \tag{42} \end{eqnarray} Let \begin{eqnarray}T_0(I-R^{\dag}R)z(t)=\left[ \begin{array}{c} 0 \\ u_1(t) \\ \end{array} \right], \tag{43} \end{eqnarray} where $u_1(t)=\Upsilon_{T_0}~z(t)\in~\mathbb{R}^{m-m_0(t)}$. Now, we can rewrite (42) as (16). Note that \begin{eqnarray}I-RR^{\dag} &=&(I-RR^{\dag})(I-RR^{\dag}) \\ &=&(I-RR^{\dag})T'_0(T^{-1}_0)'(I-RR^{\dag}) \\ &=&\left[ \begin{array}{cc} 0 & \Upsilon'_{T_0} \\ \end{array} \right](T^{-1}_0)'(I-RR^{\dag}), \tag{44} \end{eqnarray} where we have used (14) to derive the last equality. By using the definitions below (14), we can rewrite (42) as \begin{eqnarray}0&=&\Upsilon'_{T_0}\Big[C_0(t)x(t)+B_0'\Theta(t)\Big]. \tag{45} \end{eqnarray} Note that $\Upsilon'_{T_0}$ is of full column rank, and thus Eq. (45) can be directly rewritten as (16).

Third, we derive the dynamics of $\Theta(t)$. Substituting (41) into (39) and using (8) yields \begin{eqnarray}0&=&\dot{P}(t)x(t)+P(t)Ax(t) +A'P(t)x(t)+A'\Theta(t)+Qx(t)+\dot{\Theta}(t) \\ & &-P(t)BR^{\dag}\left(B'P(t)x(t)+B'\Theta(t)\right)+P(t)B(I-R^{\dag}R)z(t) \\ &=&\dot{\Theta}(t)+\left(A'-P(t)BR^{\dag}B'\right)\Theta(t)+P(t)B(I-R^{\dag}R)z(t). \tag{46} \end{eqnarray} As $(I-R^{\dag}R)^2=I-R^{\dag}R$, we find that \begin{eqnarray}P(t)B(I-R^{\dag}R)z(t) &=&P(t)B(I-R^{\dag}R)T_0^{-1}T_0(I-R^{\dag}R)z(t) \\ &=&P(t)B(I-R^{\dag}R)T_0^{-1}\left[ \begin{array}{c} 0 \\ u_1(t) \\ \end{array} \right] \\ &=&\left[ \begin{array}{cc} * & C_0'(t) \\ \end{array} \right]\left[ \begin{array}{c} 0 \\ u_1(t) \\ \end{array} \right]=C_0'(t)u_1(t). \tag{47} \end{eqnarray} Thus, from (46) we obtain $\dot{\Theta}(t)=-[A'_0(t)\Theta(t)+C'_0(t)~u_1(t)],$ which implies that the dynamics of $\Theta(t)$ is given by (18).

Finally, we derive the dynamics equation (17). By substituting (41) into (5) and combining this with the fact that $B(I-R^{\dag}R)z(t)=B_0u_1(t)$, which can be obtained in a similar way to (47), we can derive the state dynamics (17).

Proof of sufficiency. Now, we show Problem 1 is solvable if there exists a $u_1(t)$ that enables us to achieve (16). In fact, if Eq. (16) is true then Eqs. (41) and (42) can be jointly rewritten as (40). Further, by reversing the process for (36)–(40), we can easily verify that $p(t)=P(t)x(t)+\Theta(t)$, where $x(t)$ and $\Theta(t)$ satisfy (16)–(18), solving (5)–(7). Thus, Problem 1 is solvable, completing the proof.

Proof of Theorem 3.3

Proof of sufficiency. Based on Lemma 3.2, it is sufficient to verify that $(\Theta(t),x(t))=(P_1(t)x(t),x(t))$ is the solution to the FBDEs (16)–(18). Taking the derivative of $P_1(t)x(t)$ yields \begin{eqnarray}\frac{{\rm d}[P_1(t)x(t)]}{{\rm d}t} &=&\dot{P}_1(t)x(t)+P_1(t)[A_0(t)+D_0P_1(t)] x(t)+P_1(t)B_0u_1(t) \\ &=&-A_0'(t)P_1(t)x(t)+P_1(t)B_0u_1(t) \\ &=&-A_0'(t)P_1(t)x(t)-C_0'(t)u_1(t), \tag{48} \end{eqnarray} where we have used (15) and (19) to derive the last equality. In addition, again using (19), we have \begin{eqnarray}C_0(t)x(t)+B_0'P_1(t)x(t)=0. \tag{49} \end{eqnarray} By comparing (16)–(18) with (49), (48), and (21), we can see that Eqs. (16)–(18) are solvable with $\Theta(t)=P_1(t)x(t)$ if $P_1(T)x(T)=0$. Thus, based on Lemma 3.2, Problem 1 is solvable.

Proof of necessity. This proof is divided into two parts. First, we consider the case where the optimal solution is of closed-loop form, namely $u_1(t)=K_1(t)x(t)$. Based on Lemma 3.2, Eqs. (16)–(18) are solvable if Problem 1 is solvable. By substituting $u_1(t)=K_1(t)x(t)$ into (17) and (18), we obtain \begin{eqnarray}\dot{x}(t)&=&A_0(t)x(t)+D_0\Theta(t)+B_{0}K_1(t)x(t), \\ \dot{\Theta}(t)&=&-\left(A'_0(t)\Theta(t)+C'_0(t)K_1(t)x(t)\right). \end{eqnarray} Solving the above FBDEs gives us $\Theta(t)=\bar{P}(t)x(t)$, where $\bar{P}(t)$ satisfies \begin{eqnarray}0=\dot{\bar{P}}(t)+\bar{P}(t)A_0(t)+\bar{P}(t)D_0\bar{P}(t)+A_0'(t)\bar{P}(t)+\left(\bar{P}(t)B_0+C_0'(t)\right)K_1(t). \tag{50} \end{eqnarray} In addition, substituting $\Theta(t)=\bar{P}(t)x(t)$ into (16) yields \begin{eqnarray}0=C_0(t)+B_0'\bar{P}(t). \tag{51} \end{eqnarray} Thus, we can reformulate (50) as \begin{eqnarray}0=\dot{\bar{P}}(t)+\bar{P}(t)A_0(t)+\bar{P}(t)D_0\bar{P}(t)+A_0'(t)\bar{P}(t). \end{eqnarray} Comparing this with (15), we find that $\bar{P}(t)=P_1(t)$. Thus, Eq. (19) follows from (51) and Eq. (20) follows from $\Theta(T)=0$ and $\Theta(T)=P_1(T)x(T)$.

Second, the case where the controller $u_1(t)$ is of open-loop form can be solved similarly to the closed-loop case. This completes the proof.

Proof of Theorem 3.5

Proof of sufficiency. Under the condition (19), it is sufficient to verify that $P_1(T)x(T)=0$, given Theorem 3.3. To do this, we first state a formula relating $P_1(t)x(t)$ to the control $u_1(t)$ in terms of its dynamics. Similar to (48), the dynamics of $P_1(t)x(t)$ is given by \begin{eqnarray}\frac{{\rm d}[P_1(t)x(t)]}{{\rm d}t}=-A_0'(t)P_1(t)x(t)-C_0'(t)u_1(t). \end{eqnarray} Solving this differential equation yields \begin{eqnarray}P_1(t)x(t) &=&\int_{t}^TP_2(t,s)C_0'(s)u_1(s){\rm d}s+P_2(t,T)C, \tag{52} \end{eqnarray} where $C=P_1(T)x(T)$.

Next, we aim to prove that $C=0$ under the controller $u_1(t)$ defined in (25). If Eq. (23) holds, then for any $x_0$, there exists a $\zeta$ such that $P_1(t_0)x_0=G_1[t_0,T]\zeta,~$ where $\zeta=G_1^{\dag}[t_0,T]P_1(t_0)x_0$. We can now rewrite $u_1(t)$ in (25) as $u_1(t)=C_0(t)P_2'(t_0,t)\zeta$. By substituting $u_1(t)$ into (52), we obtain \begin{eqnarray}P_1(t_0)x_0 =\left[\int_{t_0}^TP_2(t_0,s)C_0'(s)C_0(s)P_2'(t_0,s){\rm d}s\right]\zeta+P_2(t_0,T)C=P_1(t_0)x_0+P_2(t_0,T)C. \end{eqnarray} As $P_2(t_0,T)$ is invertible, we have $C=0$, implying that $P_1(T)x(T)=0$. This completes the proof of sufficiency based on Theorem 3.3.

Proof of necessity. If the control problem is solvable, it follows from Theorem 3.3 that there exists a $P_1(t)$ such that Eq. (19) holds. We now prove that Eq. (23) does indeed hold. Otherwise, we would have that ${\rm~Range}~\big[P_1(t_0)\big]\nsubseteq {\rm~Range}~\left(G_1[t_0,T]\right)$, meaning that a non-zero vector $\rho$ would exist such that $\rho'P_1(t_0)\rho\neq0,\rho'G_1[t_0,T]\rho=0$. Then, we would obtain \begin{eqnarray}0=\rho'G_1[t_0,T]\rho=\rho'\left[\int_{t_0}^TP_2(t_0,s)C_0'(s)C_0(s)P_2'(t_0,s){\rm d}s\right]\rho =\int_{t_0}^T\|C_0(s)P_2'(t_0,s)\rho\|^2{\rm d}s, \end{eqnarray} implying that $C_0(s)P_2'(t_0,s)\rho=0$. Thus, we would have $\rho'\int_{t_0}^TP_2(t_0,s)C_0'(s)u_1(s){\rm~d}s=0$. Let $x_0=\rho$. From $\Theta(t_0)=P_1(t_0)x_0$, we would then have $\Theta(t_0)=P_1(t_0)\rho$. Combining this with $\Theta(t)=\int_{t}^TP_2(t,s)C_0'(s)u_1(s){\rm~d}s$ gives $$\rho'P_1(t_0)\rho=\rho'\left[\int_{t}^TP_2(t,s)C_0'(s)u_1(s){\rm d}s\right]\rho=0.$$ This is a contradiction, so Eq. (23) must hold, completing the proof.

Proof of Theorem 3.6

Let \begin{eqnarray}y(t)=\mathcal{T}_1'(t)x(t)=\left[ \begin{array}{c} y_1(t) \\ y_2(t) \\ \end{array} \right]. \tag{53} \end{eqnarray} Then, using (17) and the feedback controller $u_1(t)=K(t)x(t)$, we have \begin{eqnarray}\dot{y}(t)&=&\dot{\mathcal{T}}_1'(t)x(t)+\mathcal{T}_1'(t)\left(A_0(t)x(t)+D_0\Theta(t)+B_0K(t)x(t)\right) \\ &=&[\dot{\mathcal{T}}_1'(t)\mathcal{T}_1(t)+\mathcal{T}_1'(t)\left(A_0(t)+D_0P_1(t)+B_0K(t)\right)\mathcal{T}_1(t)]\mathcal{T}_1'(t)x(t) \\ &=&\left(\left[ \begin{array}{c} \tilde{T}_{1}(t) \\ \tilde{T}_{2}(t) \\ \end{array} \right]+\left[ \begin{array}{c} \hat{A}_{1}(t) \\ \hat{A}_{2}(t) \\ \end{array} \right]+\left[ \begin{array}{c} B_{1}(t) \\ B_{2}(t) \\ \end{array} \right]\mathcal{T}_1'(t) K(t)\mathcal{T}_1(t)\right)y(t) \\ &=&\left[ \begin{array}{c} \tilde{T}_{1}(t)+\hat{A}_{1}(t)+B_{1}(t)\mathcal{T}_1'(t)K(t)\mathcal{T}_1(t) \\ \tilde{T}_{2}(t)+\hat{A}_{2}(t)+B_{2}(t)\mathcal{T}_1'(t)K(t)\mathcal{T}_1(t) \\ \end{array} \right]y(t). \end{eqnarray} By applying (26), we obtain $$\left[ \begin{array}{c} \dot{y}_1(t) \\ \dot{y}_2(t) \\ \end{array} \right]=\left[ \begin{array}{cc} \frac{I}{t-T} & 0 \\ * & * \\ \end{array} \right]\left[ \begin{array}{c} y_1(t) \\ y_2(t) \\ \end{array} \right].$$ This implies that $\dot{y}_1(t)=\frac{I}{t-T}y_1(t)$. Then, solving this equation gives us $y_1(t)=\frac{T-t}{T-t_0}y_1(t_0)$, further implying that \begin{eqnarray}y_1(T)=0. \tag{54} \end{eqnarray} As \begin{eqnarray}0=\mathcal{T}_1'(T)P_1(T)\mathcal{T}_1(T)\mathcal{T}_1'(T)x(T)=\left[ \begin{array}{cc} \hat{P}(T) & 0 \\ 0 & 0 \\ \end{array} \right]y(T)=\hat{P}(T)y_1(T), \tag{55} \end{eqnarray} we can combine this with the invertibility of $\hat{P}(t)$ to obtain $y_1(T)=0$. We also have (53) and $y(T)=\mathcal{T}_1'(T)x(T)={\tiny[ ~~~~~~~~~~~~~~~~~~~~~~~~~~~\begin{array}{c} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y_2(T)~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array} ~~~~~~~~~~~~~~~~~~~~~~~~~]}$, so that \begin{eqnarray}P_1(T)x(T)&=&P_1(T)\mathcal{T}_1(T) \left[ \begin{array}{c} 0 \\ y_2(T) \\ \end{array} \right] =\mathcal{T}_1(T)\mathcal{T}'_1(T) P_1(T)\mathcal{T}_1(T) \left[ \begin{array}{c} 0 \\ y_2(T) \\ \end{array} \right] \\ &=&\mathcal{T}_1(T) \left[ \begin{array}{cc} \hat{P}(T) & 0 \\ 0 & 0 \\ \end{array} \right] \left[ \begin{array}{c} 0 \\ y_2(T) \\ \end{array} \right] =0. \end{eqnarray} This completes the proof.

Proof of Theorem 4.3

Proof of sufficiency. As solutions $P$ and $P_1$ exist to (27) and (32) that satisfy $P+P_1\geq~0$, we will show that the cost function is bounded below by $x_0'(P+P_1)x_0$. By taking the derivative of $x'(t)(P+P_1)x(t)$, we obtain \begin{eqnarray}\frac{\rm d}{{\rm d}t}\left(x'(t)(P+P_1)x(t)\right) &=&\left(Ax(t)+Bu(t)\right)'(P+P_1)x(t)+x'(t)(P+P_1)\left(Ax(t)+Bu(t)\right) \\ &=&-x'(t)Qx(t)+x'(t)(P+P_1)BR^{\dag}B'(P+P_1)x(t)+u'(t)B'(P+P_1)x(t) \\ & &+x'(t)(P+P_1)Bu(t), \end{eqnarray} where we have used (27) and (32) to derive the last equality. By integrating this from $0$ to $T$, we can further obtain \begin{eqnarray}\int_{0}^T& &\!\!\!\!\left(x'(t)Qx(t)+u'(t)Ru(t)\right){\rm d}t \\ & &=x'(0)(P+P_1)x(0)-x'(T)(P+P_1)x(T)+\int_0^T\bigg(u'(t)Ru(t)+x'(t)(P+P_1)BR^{\dag}B'(P+P_1)x(t) \\ & & +u'(t)B'(P+P_1)x(t)+x'(t)(P+P_1)Bu(t)\bigg){\rm d}t \\ & &=x'(0)(P+P_1)x(0)-x'(T)(P+P_1)x(T)+\int_0^T\Big[\left(u(t)+R^{\dag}B'(P+P_1)x(t)\right)'R(u(t) \\ & & +R^{\dag}B'(P+P_1)x(t))+u'(t)(I-RR^{\dag})B'(P+P_1)x(t)+x'(t)(P+P_1)B(I-R^{\dag}R)u(t)\Big]{\rm d}t \\ & &=x'(0)(P+P_1)x(0)-x'(T)(P+P_1)x(T)+\int_0^T\left(u(t)+R^{\dag}B'(P+P_1)x(t)\right)'R(u(t) \\ & & +R^{\dag}B'(P+P_1)x(t)){\rm d}t, \end{eqnarray} where we have used $(I-RR^{\dag})B'(P+P_1)=0$ to derive the last equality, obtained from (33). As $u(t)\in~\mathcal{U}$, we thus have $\lim_{T\rightarrow\infty}x'(T)(P+P_1)x(T)=0$. This implies that \begin{eqnarray}J(x_0;u)&=&\lim_{T\rightarrow\infty}\int_{0}^T\left(x'(t)Qx(t)+u'(t)Ru(t)\right){\rm d}t \\ &=&x'(0)(P+P_1)x(0)+\lim_{T\rightarrow\infty}\int_0^T\left(u(t)+R^{\dag}B'(P+P_1)x(t)\right)'R\left(u(t)+R^{\dag}B'(P+P_1)x(t)\right){\rm d}t. \tag{56} \end{eqnarray} Because $R\geq~0$, we obtain $J(x_0;u)\geq~x'(0)(P+P_1)x(0)$.

Next, we show that the controller (35) is stabilizing. Substituting (35) into (1) yields \begin{eqnarray}\dot{x}(t)&=&Ax(t)-BR^{\dag}B'(P+P_1)x(t)+BG_0Kx(t) \\ &=&(A_0+D_0P_1)x(t)+BG_0Kx(t) \\ &=&(A_0+D_0P_1)x(t)+BT_0^{-1}\left[ \begin{array}{c} 0 \\ K \\ \end{array} \right]x(t). \tag{57} \end{eqnarray} Because $\Upsilon_{T_0}$ is of full row rank, there exists a $K_1$ such that $\Upsilon_{T_0}K_1=K$. From (14), we find that \begin{eqnarray}T_0(I-R^{\dag}R)K_1=\left[ \begin{array}{c} 0 \\ \Upsilon_{T_0} \\ \end{array} \right]K_1=\left[ \begin{array}{c} 0 \\ K \\ \end{array} \right]. \tag{58} \end{eqnarray} By substituting the above equation into (57), we then have \begin{eqnarray}\dot{x}(t)&=&(A_0+D_0P_1)x(t)+BT_0^{-1}T_0(I-R^{\dag}R)K_1x(t) \\ &=&(A_0+D_0P_1)x(t)+B(I-R^{\dag}R)T_0^{-1}T_0(I-R^{\dag}R)K_1x(t) \\ &=&(A_0+D_0P_1)x(t)+B_0\Upsilon_{T_0}K_1x(t) \\ &=&(A_0+D_0P_1)x(t)+B_0Kx(t). \end{eqnarray} $K$ was chosen to satisfy that $A_0+D_0P_1+B_0K$ is stable, so the above system is stable, and hence the controller (35) is stabilizing.

Finally, we substitute the stabilizing controller (35) into the cost function (56) to verify that Eq. (35) is an optimal controller as desired. In fact, with this controller, Eq. (56) becomes \begin{eqnarray}J(x_0;u) &=&x'(0)(P+P_1)x(0)+\lim_{T\rightarrow\infty}\int_0^T\left([-R^{\dag}B'(P+P_1)+G_0K]x(t)+R^{\dag}B'(P+P_1)x(t)\right)' \\ & &\times R\left([-R^{\dag}B'(P+P_1)+G_0K]x(t)+R^{\dag}B'(P+P_1)x(t)\right){\rm d}t \\ &=&x'(0)(P+P_1)x(0)+\lim_{T\rightarrow\infty}\int_0^Tx'(t)K'G_0'RG_0Kx(t){\rm d}t. \tag{59} \end{eqnarray} By again using (58), it follows that $G_0K=T_0^{-1}{\tiny[ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\begin{array}{c} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~K~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~]}=(I-R^{\dag}R)K_1$. This implies that $RG_0K=R(I-R^{\dag}R)K_1=0$. Thus, with the controller (35), the cost function (59) reduces to \begin{eqnarray}J(x_0;u)=x'(0)(P+P_1)x(0). \end{eqnarray} This shows that Eq. (35) is an optimal controller, and the optimal cost is $J^*(x_0;u)=x_0'(P+P_1)x_0$.

Proof of necessity. Here, we derive the three conditions given in the theorem. First, we discuss the results for the finite-horizon optimization problem. Considering the asymptotic behavior of the solutions to the Riccati differential equations enables us to obtain the first and second conditions. Then, by applying the maximum principle, we find that the stabilizability condition is as stated by the third condition. The detailed proof is given below.

First, based on Theorem 3.3, there exists a $P^T(t)$ in (8) and a $P_1^T(t)$ in (15) with terminal values of $P^T(T)=0$ and $P_1^T(T)$ such that Eq. (19) holds, and there also exists a $u_{1}(t)$ that achieves (20), where $x(t)$ obeys (21) with initial value $x(0)=x_0$. In this case, the optimal cost is given by \begin{eqnarray}J_T^*(x_0;u)=x_0'\hat{P}^T(0)x_0, \tag{60} \end{eqnarray} where $\hat{P}^T(t)=P^T(t)+P_1^T(t)$. Given that $Q\geq0$ and $R\geq~0$, we have $J_T(x_0;u)\geq0$. Accordingly, for $T_1\leq~T_2$, we obtain $J_{T_1}(x_0;u)\leq~J_{T_2}(x_0;u)$. Together with (60) and the arbitrariness of $x_0$, we thus find that \begin{eqnarray}\hat{P}^{T_1}(0)\leq \hat{P}^{T_2}(0). \tag{61} \end{eqnarray} In addition, consider the cost function \begin{eqnarray}J_T^t(x_0;u)=\int_{t}^T[x'(t)Qx(t)+u'(t)Ru(t)]{\rm d}t. \end{eqnarray} By applying a similar argument to that for Theorem 3.3, the optimal cost yielded by minimizing $J_T^t(x_0;u)$ subject to (1) is given by \begin{eqnarray}J_T^t(x_0;u)=x'(t)\hat{P}^{T}(t)x(t). \end{eqnarray} For $t_1\leq~t_2$, we have that \begin{eqnarray}J_T^{t_1}(x_0;u)\geq J_T^{t_2}(x_0;u), \end{eqnarray} which implies that \begin{eqnarray}\hat{P}^{T}(t_1)\geq \hat{P}^{T}(t_2). \tag{62} \end{eqnarray} Combining (61) and (62), we see that $\hat{P}^T(t)$ is non-decreasing with respect to $T$ and that $\hat{P}^T(t)$ is non-increasing with respect to $t$.

Next, we show the uniform boundedness of $\hat{P}^T(t)$. As there exists an optimal and stabilizing controller, there also exists a positive constant $c$ such that \begin{eqnarray}J_T^t(x_0;u)&\leq& \int_0^\infty\left(x'(t)Qx(t)+u'(t)Ru(t)\right){\rm d}t\leq c\|x_0\|^2. \end{eqnarray} Combining this with (60), it follows that $\hat{P}^T(t)\leq~cI.$ As all the system matrices are time-invariant, $\hat{P}^T(t)$ is also time-invariant, i.e., $\hat{P}^T(t)=\hat{P}^{T-t}(0).$ Recalling (61) and (62), this shows that the limit $\lim_{T\rightarrow\infty}~\hat{P}^T(t)=\hat{P}$ exists. Moreover, by letting $t\rightarrow\infty$ in $\hat{P}^T(t)=P^T(t)+P_1^T(t)$, we see that $\hat{P}$ satisfies \begin{eqnarray}0&=&A'\hat{P}+\hat{P}A+Q-\hat{P}BR^{\dag}B'\hat{P}. \end{eqnarray} This is exactly the same equation for $P$; hence, Eq. (27) is solvable. This further implies that Eq. (32) admits a solution $P_1$ and that $\hat{P}=P+P_1$. Likewise, letting $t\rightarrow\infty$ in (19) yields $C_0+B_0'P_1=0,$ which is exactly (33).

Finally, by applying the maximum principle, the optimal solution satisfies \begin{eqnarray}& &\dot{x}(t)=Ax(t)+Bu(t), \tag{63} \\ & &\dot{p}(t)=-A'p(t)-Qx(t), \tag{64} \\ & &0=Ru(t)+B'p(t), \tag{65} \end{eqnarray} with $\lim_{t\rightarrow\infty}p(t)=0$ and $x(0)=x_0$. Recalling that the optimal solution is also stabilizing, we obtain \begin{eqnarray}\lim_{t\rightarrow\infty}x(t)=0, \end{eqnarray} and hence that \begin{eqnarray}\lim_{t\rightarrow\infty}Px(t)=0, \tag{66} \\ \lim_{t\rightarrow\infty}P_1x(t)=0. \tag{67} \end{eqnarray} Let \begin{eqnarray}p(t)=Px(t)+\Theta(t), \tag{68} \end{eqnarray} where $P(t)$ obeys (27) and $\Theta(t)$ is to be determined. From (66) and $\lim_{t\rightarrow\infty}p(t)=0$, we then have $\lim_{t\rightarrow\infty}\Theta(t)=0$.

By substituting (68) into (65), we obtain \begin{eqnarray}0&=&Ru(t)+B'Px(t)+B'\Theta(t). \end{eqnarray} This implies that \begin{eqnarray}u(t)&=&-R^{\dag}\left(B'Px(t)+B'\Theta(t)\right)+(I-R^{\dag}R)z(t) \\ \tag{69} \end{eqnarray} and \begin{eqnarray}C_0x(t)+B_0'\Theta(t)=0, \tag{70} \end{eqnarray} where $z(t)$ is an arbitrary vector of compatible dimension.

Substituting (69) into (1) reduces the state dynamics to \begin{eqnarray}\dot{x}(t)&=&Ax(t)-BR^{\dag}\left(B'Px(t)+B'\Theta(t)\right)+B(I-R^{\dag}R)z(t) \\ &=&A_0x(t)+D_0\Theta(t)+B_0u_1(t), \tag{71} \end{eqnarray} where we have used $T_0(I-R^{\dag}R)z(t)={\tiny[ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\begin{array}{c} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Upsilon_{T_0}~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~]}z(t)\triangleq{\tiny[ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\begin{array}{c} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~u_1(t)~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~]} $ in the derivation of the last equality. Taking the derivative of (68) yields \begin{eqnarray}\dot{p}(t)=P\dot{x}(t)+\dot{\Theta}(t)=P\left(A_0x(t)+D_0\Theta(t)+B_0u_1(t)\right)+\dot{\Theta}(t). \end{eqnarray} Comparing this with (64) and using (32), we have \begin{eqnarray}\dot{\Theta}(t)&=&-A_0'\Theta(t)-C_0'u_1(t). \tag{72} \end{eqnarray} We now prove that the solution to the FBDEs (70)–(72) is $\Theta(t)=P_1x(t)$, where $x(t)$ satisfies (34). By taking the derivative of $P_1x(t)$, we obtain \begin{eqnarray}\frac{\rm d}{{\rm d}t}\left(P_1x(t)\right) &=&P_1(A_0+D_0P_1)x(t)+P_1B_0u_1(t) \\ &=&-A_0'P_1x(t)+P_1B_0u_1(t) \\ &=&-A_0'P_1x(t)-C_0'u_1(t), \tag{73} \end{eqnarray} where we have used (33) to derive the last equality. By comparing (34), (73), and (33) with (70)–(72), we immediately obtain \begin{eqnarray}\Theta(t)=P_1x(t). \end{eqnarray} Accordingly, the state dynamics is given by (34). To ensure the state's stability, the state dynamics must be stabilizable, giving us the third condition. This completes the proof.


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