By S.I. Lyashko
The writer of this e-book made an try and create the final concept of optimization of linear platforms (both allotted and lumped) with a novel keep an eye on. The ebook touches upon quite a lot of concerns comparable to solvability of boundary values difficulties for partial differential equations with generalized right-hand facets, the lifestyles of optimum controls, the required stipulations of optimality, the controllability of structures, numerical tools of approximation of generalized recommendations of preliminary boundary price issues of generalized information, and numerical tools for approximation of optimum controls. particularly, the issues of optimization of linear platforms with lumped controls (pulse, element, pointwise, cellular and so forth) are investigated intimately.
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Extra resources for Generalized Optimal Control of Linear Systems with Distributed Parameters (Applied Optimization)
12) under the following assumptions: 1) the performance criterion is a functional which is weakly lower semicontinuous with respect to the system state u(t,x,h) and below bounded; OPTIMIZATION OF LINEAR SYSTEMS ... 2) and (24) are valid. 12). Similarly we can prove the following T h e o r e m 1 2 . 12). Note, that we may consider the problem of optimal control when the right-hand side of the state equation is a linear combination of the functionals Consider, for example, the problem 42 Chapter 1 where The control is By we shall denote a functional defined on smooth in following way: Suppose that functions in the and and let As far as are negative spaces constructed on the pairs similarly to the cases and and we shall prove that the right-hand side of (27) belongs to the space and hence, to the Let us prove that mapping.
Consider the application of these theorems in the case of the optimization of distributed systems with point controls. The urgency of such investigations is stipulated both by the development of new technologies and by the simplicity of the control realization. In view of specific character of control in these problems it is possible to obtain more interesting results [80-86] Let the studied system be described by the linear partial differential equation Consider the optimal control problems for the systems, which are described by the equation (3) with right-hand sides in the following forms: OPTIMIZATION OF LINEAR SYSTEMS ...
3 guarantees that the function u(t,x,h) belongs to the space 46 Chapter 2 Suppose that and moreover this imbedding is continuous. T h e o r e m 1. Provides that conditions above mentioned are satisfied, the functional (1) is differentiable by Gâteaux in the space and its gradient is of the form: where is a control, v(t, x) is a solution of the problem Proof. 12) corresponding to these controls. Denote Then the increment of the performance criterion may be represented in the following form Define the adjoint state as a solution of the problem GENERAL PRINCIPLES OF INVESTIGATION ...