Abstract
The present paper deals with a class of second-order PDE constrained controlled optimization problems with application in Lagrange–Hamilton dynamics. Concretely, we formulate and prove necessary conditions of optimality for the considered class of control problems driven by multiple integral cost functionals involving second-order partial derivatives. Moreover, an illustrative example is provided to highlight the effectiveness of the results derived in the paper. In the final part of the paper, we present an algorithm to summarize the steps for solving a control problem such as the one investigated here.
Highlights
Calculus of Variations and Optimal Control Theory are two mathematical fields with strong and important connections, with significant applications in applied sciences, engineering, data analysis, and classification
Treanţă [13] has focused on the optimization of some simple, multiple or curvilinear integral functionals subject to ODE, PDE or isoperimetric constraints
In the present paper, inspired by the previous research in this field, we investigate a new class of PDE constrained optimization problems driven by multiple integral cost functionals which involve second-order partial derivatives
Summary
Calculus of Variations and Optimal Control Theory are two mathematical fields with strong and important connections, with significant applications in applied sciences, engineering, data analysis, and classification. By using the Pontryagin’s principle, Schmitendorf [22] studied the necessary conditions of optimality associated with a class of control problems involving isoperimetric and inequality constraints at the terminal time. In the present paper, inspired by the previous research in this field, we investigate a new class of PDE constrained optimization problems driven by multiple integral cost functionals which involve second-order partial derivatives. The mathematical framework developed in this paper is more general than in Hestenes [2], Schmitendorf [22], Udrişte and Ţevy [4], or Treanţă [13], both by the presence of controlled multiple integrals and by the inclusion of the new proof associated with the main result and the second-order partial derivatives.
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