Existence theorems for optimization problems concerning hyperbolic partial differential equations

Abstract

Existence theorems are given for optimal control problems with partial integro-differential equations of hyperbolic type. A variety of conditions are given depending on whether (i) the cost functional is in the Lagrange form or Mayer form, (ii) the controls vary in a fixed compact set or variable but closed sets, (iii) the required solutions are generalized or usual, (iv) the defining equations are nonlinear or linear in the state variable or linear in the control, (v) the trajectories belong to the usual Sobolev classW p 2 ,p>1, or the special Sobolev classW p * , 1⩽p⩽∞. Several examples are given to illustrate the above cases. Necessary conditions for some of these problems have been discussed in an earlier paper.