Existence Theorems For Optimal Problems Research Articles

Existence theorems for optimization problems concerning linear, hyperbolic partial differential equations without convexity conditions

Integral representations are obtained for solutions of a Darboux problem in a rectangle and used to prove Neustadt-type existence theorems for optimal control problems with trajectories satisfying linear, hyperbolic partial differential equations with Darboux-type boundary data. The proof bears on the fact that, in this situation, for each generalized solution, there is a usual solution where the functional takes the same value.

Existence theorems for optimization problems concerning hyperbolic partial differential equations

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.

Existence Theorems for Optimal Problems with Vector-Valued Cost Function

Existence Theorems for Optimal Problems with Vector-Valued Cost Function

Existence theorems for optimal problems with vector-valued cost function

Abstract : The paper considers the following optimal control problem. Consider a control system (0.1) y dot = f(t,y,u) where f maps J sub o X Y X E into Y, J sub o = (a sub o, b sub o) is an interval, and Y,E are Euclidean spaces. By a solution of (0.1) is meant a triple (J,y,u), where J included in J sub o is an interval, y: J approaches Y is absolutely continuous, u: J approaches E is Lebesgue measurable, and y dot(t) = f(t,y(t),u(t)), u(t) epsilon U(t,y(t)) almost everywhere (a.e.) in J. The control domain U is a map from J sub o X Y into subsets of E.