A survey of optimal control of distributed-parameter systems

The problem of optimal control whenever the process is described by an equation in a Banach space or other abstract space was studied in a series of earlier papers ([ 1 ] [6]). In this connection the application of the most general results ([5], [6]) t o t h e solution of concrete problems turns out to be difficult because the analytic apparatus utilized here is rather complicated and is not always accessible to a wide circle of specialists. In the present note we study an elementary procedure for obtaining optimality conditions for certain problems where the control process is described by an operator equation in a Banach space. The basis of the method is an idea of L. I. Rozonoer [7] for proving the Pontryagin maximum principle. The examples cited below show that the result obtained here is more general for a series of problems than the corresponding conclusions in the papers [1] and [5]. Thus, let B1, B2, B3 and B4 be Banach spaces with B1 C_ B.3 and.B2 C_. B3. We denote by A a linear operator, which is in general unbounded and which acts from B1 into B3. Moreover, to each pair of elements x eB2 and u ~ B4 let there correspond an elementf(x, u) ~:B3, where the ope ra to r f has a continuous Frgchet derivative Of/Ox which, together with f, is continuous in x and in u. Each element u e B4 will be called a control, and those u belonging to the set U C_ B4 will be called admissible controls. We consider the process described by the equation

Optimal control in a Banach space

Optimal control of distributed-parameter systems is studied from a function space point of view. The adjoint-space technique of Balakrishnam is applied to solve a general class of linear distributed-parameter final-value problems in Hilbert space. The minimum-time control problem with multi-norm constraints for a general class of approximately controllable systems is treated, using the technique of Sarachik and Kranc, through the application of Holder's inequality. Finally, a generalized Holder's inequality for more than two Banach spaces is given, which is useful for solving a certain class of nonlinear control problems by the same methods. Two examples are included which illustrate the theory.

Distributed-parameter control in function space

A one-dimensional heat conduction system is optimized with respect to a quadratic performance index. A sampled-data model is employed to obtain a representation of the system for which the standard form of the matrix Riccati equation and the Kalman equation may be employed. Solutions obtained for the two equations are in excellent agreement with each other and also with a published result.

A Hamilton-Jacobi approach to the optimal control of distributed parameter systems†

Chatter mode is used advantageously if the ideal model dynamics is described by a switching function, the chatter mode is reached quickly and then sustained, the chatter frequency is reduced, and any zero in the plant transfer function is cancelled.

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The use of the chatter mode in self-adaptive systems

In this paper distributed parameter systems governed by systems of linear partial differential equations have been treated. The admissibly control functions are assumed to belong to a closed ball in the appropriate Banach space. Formulae have been derived, such that the norm of the control function in the Banach space in which they are assumed to belong is minimized, such that the control becomes simultaneously time optimal.

A. K. Chaudhuri

Papers

Concerning Optimum Control of Linear Distributed Parameter Systems