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.
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