×

Geometry and optimal control. (English) Zbl 1067.49500

Baillieul, John B. (ed.) et al., Mathematical control theory. With a foreword by Sanjoy K. Mitter. Dedicated to Roger Ware Brockett on the occasion of his 60th birthday. New York, NY: Springer (ISBN 0-387-98317-1). 140-198 (1998).
Here Sussmann provides an extended discussion of the maximum principle, from its historical roots in the calculus of variations to the present. He shows how the subjects of differential geometry and variational analysis have been inseparable from their earliest days, and argues for the importance of giving the maximum principle (a) a geometric formulation concerning separation properties of the reachable set, and (b) an intrinsic formulation on manifolds. He explains how such formulations can be constructed and justified by making a detailed analysis of control systems with states in finite-dimensional Euclidean space, using a carefully-optimized revision of the “needle variations” employed in the earliest proofs. This calls for a full collection of new terminology and tools: semidifferentials, multidifferentials, weakly approximating cones, etc., whose definitions and properties are fully described. Various combinations of technical hypotheses allow various forms of the maximum principle to be established. Sussmann describes the classical setting, Clarke’s nonsmooth approach, the role of Warga’s derivate containers, and finally his own contributions, culminating in a rather general maximum principle that admits an intrinsic formulation and allows for higher-order information to be incorporated into the transversality conditions. This paper provides a helpful overview of Sussmann’s viewpoint and recent achievements, but it is not self-contained: key elements of the proof of the main results appear elsewhere, especially in another paper of his [in Optimal control (Gainesville, FL, 1997), Kluwer Acad. Publ., Dordrecht, 436–487 (1998; Zbl 0981.49013)].
For the entire collection see [Zbl 0904.00033].

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49J52 Nonsmooth analysis
58E25 Applications of variational problems to control theory
93B27 Geometric methods
93C10 Nonlinear systems in control theory

Citations:

Zbl 0981.49013
PDFBibTeX XMLCite