Asymptotic expansion of solution to singularly perturbed optimal control problem with convex integral quality functional with terminal part depending on slow and fast variables

A. R. Danilin ; A. A. Shaburov

Ufa mathematical journal, Tome 11 (2019) no. 2, pp. 82-96

Résumé

We consider an optimal control problem with a convex integral quality functional for a linear system with fast and slow variables in the class of piecewise continuous controls with smooth constraints on the control $$ \left\{ \begin{aligned} \dot{x}_{\varepsilon} = A_{11}x_{\varepsilon} + A_{12}y_{\varepsilon}+B_{1}u,\qquad t\in[0,T],\qquad \|u\|\leqslant 1,\\ \varepsilon\dot{y}_{\varepsilon} = A_{22}y_{\varepsilon} + B_{2}u,\quad x_{\varepsilon}(0)=x^{0},\qquad y_{\varepsilon}(0)=y^{0},\qquad \nabla\varphi_2(0)=0, \\ (u)\mathop{:=}\nolimits \varphi_1\left(x_\varepsilon(T)\right) + \varphi_2\left(y_\varepsilon(T)\right) + \int\limits_{0}^{T}\|u(t)\|^2\,dt\rightarrow \min, \end{aligned} \right. $$ where $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}^{m}$, $ u\in\mathbb{R}^{r}$; $A_{ij}$ and $B_{i}$, $i,j=1,2$, are constant matrices of corresponding dimension, and the functions $\varphi_{1}(\cdot), \varphi_{2}(\cdot)$ are continuously differentiable in $\mathbb{R}^{n}, \mathbb{R}^{m},$ strictly convex, and cofinite in the sense of the convex analysis. In the general case, for such problem, the Pontryagin maximum principle is a necessary and sufficient optimality condition and there exist unique vectors $l_\varepsilon$ and $\rho_\varepsilon$ determining an optimal control by the formula $$ u_{\varepsilon}(T-t):= \frac{C_{1,\varepsilon}^{*}(t)l_\varepsilon + C_{2,\varepsilon}^{*}(t)\rho_\varepsilon} {S\left(\|C_{1,\varepsilon}^{*}(t)l_\varepsilon + C_{2,\varepsilon}^{*}(t)\rho_\varepsilon\|\right)}, $$ where \begin{align*} C_{1,\varepsilon}^{*}(t):= B^*_1 e^{A^*_{11}t} + \varepsilon^{-1}B^*_2\mathcal{W^*}_\varepsilon(t),\quad C_{2,\varepsilon}^{*}(t):= \varepsilon^{-1} B^*_2 e^{A^*_{22} t/\varepsilon}, \\ \mathcal{W}_\varepsilon(t):= e^{A_{11}t}\int\limits_{0}^{t} e^{-A_{11}\tau}A_{12}e^{A_{22} \tau/\varepsilon}\,d\tau, \quad S(\xi)\mathop{:=}\nolimits \left\{ \begin{aligned} 2,\qquad 0\leqslant \xi\leqslant2, \\ \xi, \qquad \xi>2. \end{aligned} \right. \end{align*} The main difference of our problem from the previous papers is that the terminal part of quality functional depends on the slow and fast variables and the controlled system is a more general form. We prove that in the case of a finite number of control change points, a power asymptotic expansion can be constructed for the initial vector of dual state $\lambda_\varepsilon=\left(l_\varepsilon^*\: \rho_\varepsilon^*\right)^*$, which determines the type of the optimal control.

Détail

Keywords: optimal control, singularly perturbed problems, asymptotic expansion, small parameter.