Asymptotic solution of the singularly perturbed infinite dimensional Riccati equation
The main problem discussed by the author is the singularly perturbed linear-quadratic control posed in a Hilbert space setting. The author notes that if the corresponding operators are bounded then the problem can be modeled by Barbashin's type integro-differential equations. He considers a linear system: \[ \begin{aligned} dx/dt & = A_1(t)x+ A_2(t)y+ B_1(t)u,\quad x(0)= x^0,\\ \varepsilon dy/dt & = A_3(t)x+ A_4(t)y+ B_2(t)u,\quad y(0)= y^0,\quad t\in[0, T].\end{aligned} \] Here, \(x\), \(y\) are state variables in Hilbert spaces \(H_x\) and \(H_y\), respectively, \(u\) is the control in the Hilbert space \(H_u\), and \(\varepsilon> 0\) is a ``small'' parameter. The reviewer is somewhat troubled by author's identification of the tangent space with the integral curves \((A:H_{x\to}H_x)\), etc\dots, but hopefully no harm is done in the present discussion. In the same spirit, the author introduces the Hilbert space \(H_z\), which is the direct sum of mutually orthogonal spaces \(H_x\) and \(H_y\). The resulting equations are \(dz/dt= A(t,\varepsilon)z+ B(t,\varepsilon)u\), \(z(0)= z^0\). The cost functional is \[ J= (z(T), F(z(T)))+ \int^T_0 \{\langle z,D(t)z\rangle_{H_z}+ \langle u,Ru\rangle_{H_u}\}dt, \] where \(\langle,\rangle\) denotes an inner product, \(F\) is a positive definite and selfadjoint operator: \[ F=\begin{matrix} \lceil F_1, & F_2\rceil \\ \lfloor F^*_2, & F_3\rfloor\end{matrix}. \] Then \(u\) can be represented as \(u= -R^{-1} B^*K(t,\varepsilon)z\). The operator \(K\) satisfies the differential equation: \[ dK/dt= -KA(t,\varepsilon)- A^*(t,\varepsilon)K= KS(t,\varepsilon)K- D(t), \] and \(S(t,\varepsilon)= BR^{-1}B^*\). (Here, the author does not specify what kind of derivative of the operator \(K\) is considered, even if it is easy to guess.) To avoid singularities, the author assumes that the solutions are of the form: \[ K(t,\varepsilon)=\begin{matrix} \lceil K_1,& \varepsilon K_2\rceil \\ \lfloor\varepsilon K^*_2, & \varepsilon K_3\rfloor\end{matrix}. \] The equations for the operator \(K\) are transformed to an easier form of three equations in a new matrix variable, and asymptotic solutions are sought for small value of \(\varepsilon\). This result takes about twenty pages to derive. This is a quite involved process, and following some rather difficult calculations, the author restricts the asymptotic solution to the first-order. Under certain hypothesis, he proves existence of an optimal control.
singularly perturbed linear-quadratic control
scientific article; zbMATH DE number 1115502
scientific article