Article
Ufa Mathematical Journal
Volume 11, Number 2, pp. 82-96
Asymptotic expansion of solution
to singularly perturbed optimal control problem with convex
integral quality functional with terminal part depending on slow
and fast variables
Danilin A.R., Shaburov A.A.
DOI:10.13108/2019-11-2-82
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Article on MathNetAbstact
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
$$
\left\{
\begin{aligned}
& \dot{x}_{\varepsilon} = A_{11}x_{\varepsilon} +
A_{12}y_{\varepsilon}+B_{1}u,\qquad
t\in[0,T],\qquad \|u\|\leqslant 1,\\
&\varepsilon\dot{y}_{\varepsilon} = A_{22}y_{\varepsilon} +
B_{2}u,\quad x_{\varepsilon}(0)=x^{0},\qquad
y_{\varepsilon}(0)=y^{0},\qquad \nabla\varphi_2(0)=0,
\\
&J(u)\mathop{:=}\nolimits \varphi_1\left(x_\varepsilon(T)\right) +
\varphi_2\left(y_\varepsilon(T)\right) +
\int\limits_{0}^{T}\|u(t)\|^2\,dt\rightarrow \min,
\end{aligned}
\right.
$$
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.