By B. Jakubczyk, Witold Respondek

ISBN-10: 0585388350

ISBN-13: 9780585388359

ISBN-10: 0824790685

ISBN-13: 9780824790684

This paintings gathers very important and promising details ends up in subfields of nonlinear keep an eye on concept, formerly to be had in journals. It provides the cutting-edge of geometric equipment, their purposes optimum keep watch over, and suggestions modifications. It goals to teach how geometric keep watch over thought attracts from different mathematical fields to create its personal strong instruments.

**Read or Download Geometry of Feedback and Optimal Control (Pure and Applied Mathematics) PDF**

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Sussmann ed. , Nonlinear Controllability and Optimal Control, Marcel Dekker, New York and Basel, 1990. 1 Symplectic Methods for Optimization and Control A. Agmchev and R. , U1. Gubkina 8, Moscow GSP-1, 117966, Russia Abstract. The goal of this paper is to demonstrate the power and productivity of the symplectic approach to optimization problems and to draw attention of specialists interested in Control Theory to this promising direction of investigations. We start from the classical problem of conditional extremum and then turn to the optimal control.

We write q > 0 (< 0) and we say that the form q is positive (negative), if q(e) > 0 (< 0) for nonzero e E E. If we substitute symbols > (<) by 2 ( S ) we obtain nonnegative (nonpositive) forms. 3. The remainder of this section is devoted to some initial facts from Symplectic geometry; for details we recommend [13]. Let C be a finite-dimensional real vector space. e. a bilinear mapping Q : C x C + R such that o(z1,zz) = -Q(Zz,Z1) VZl,Z2 E C, and the relation ~ ( zz'), = 0 Vz' E C implies z = 0. The space C with a given symplectic form Q on it is called symplectic.

Let e l , , . , ,e,, f l , , . ,f n be a canonical basis in C and let us write z = (xiei + tifa). The Hamiltonian system takes the standard form in coordinates xi, ti: cy=, 4. Let N be a 2n-dimensional smooth manifold. A smooth nondegene- rate closed differential 2-form cr on N is called a symplectic structure on this manifold. The nondegeneracy of the form : y + uy,y E N , means that cry is a symplectic form on the tangent space TUNVy E N ; “closed” means that da = 0, where d is the exterior differential.

### Geometry of Feedback and Optimal Control (Pure and Applied Mathematics) by B. Jakubczyk, Witold Respondek

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