Introduction to Optimal Control for Systems with Distributed Parameters. I. Frechet Differentiability in Optimal Control of Parabolic Pdes – Part 1
چکیده
Today I start a series of lectures on the optimal control of systems with distributed parameters, that is to say, optimal control of systems described by partial differential equations. Optimal Control theory is the generalization of the classical calculus of variations where minimization of the functional is pursued in the class of non-smooth functions. In XX century optimal control theory made a huge advance and in particular optimal control of systems described by ordinary differential equations has a complete theory and Pontryagin’s Maximum Principle is a major result which generalizes classical Euler-Lagrange condition to the context of the optimal control problem with the control set in the space of measurable and Lebesgue integrable functions. Optimal Control of infinite dimensional systems described by the PDEs has also made a tremendous advance. However, it is not complete as its finite dimensional counterpart. The goal of this series of lectures to derive Frechet gradient and formulate optimality conditions for the optimal control problem for the system described by the second order parabolic PDE. Today I am going to consider optimal control problem for the system described by the second order parabolic PDE. I refer to the reference [1]. Let me describe the problem. Minimize the functional