Quadratic Performance Criteria in Boundary Control of Linear Symmetric Hyperbolic Systems

Abstract

The present article develops necessary and sufficient conditions for the solution of the following optimal control problem. We let $w(x,t)$, $u(t)$ solve the hyperbolic mixed initial-boundary value problem \[ w(x,0) = w_0 (x),\]\[ \frac{{\partial w}} {{\partial t}} = A(x)\frac{{\partial w}} {{\partial x}} + B(x)w, \]\[ C_0 w(0,t) \equiv 0,\quad C_1 w(1,t) \equiv Cu(t) \] in the region $0 \leqq x \leqq 1$, $t \geqq 0$. For $t_1 > 0$ we seek to minimize a quadratic cost \[ J(w_0 ,u,t_1 ) = \int_0^{t_1 } {\left[\int_0^1 {\int_0^1 {w(x,t)^T W(x,\xi )w(\xi ,t)dxd\xi + u(t)^T Uu(t)} } \right]dt} \]\[ + \int_0^1 {\int_0^1 {w(x,t_1 )^T G(x,\xi )w(\xi ,t_1 )dxd\xi .} } \] In addition to the development of optimality conditions, the paper also presents a synthesis of the optimal solution in terms of a matrix Riccati partial differential equation. The case $t_1 = + \infty $ is also discussed.