Exploiting Characteristics in Stationary Action Problems

• Published in: Applied mathematics & optimization Vol. 84; no. Suppl 1; pp. 733 - 765
• Main Authors: Basco, Vincenzo, Dower, Peter M., McEneaney, William M., Yegorov, Ivan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Applied mathematics; Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Control; Convexity; Hilbert space; Mass-spring systems; Mathematical and Computational Physics; Mathematical functions; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Optimal control; Optimization; Partial differential equations; Principle of least action; Simulation; State feedback; Systems Theory; Theoretical; Trajectory control; Velocity; Primary 65M25; Secondary 35F21; 49S05; Characteristics; Optimal control; Hamilton–Jacobi–Bellman partial differential equations; Stationary action; 49J15
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-021-09784-6

Connections between the principle of least action and optimal control are explored with a view to describing the trajectories of energy conserving systems, subject to temporal boundary conditions, as solutions of corresponding systems of characteristics equations on arbitrary time horizons. Motivated by the relaxation of least action to stationary action for longer time horizons, due to loss of convexity of the action functional, a corresponding relaxation of optimal control problems to stationary control problems is considered. In characterizing the attendant stationary controls, corresponding to generalized velocity trajectories, an auxiliary stationary control problem is posed with respect to the characteristic system of interest. Using this auxiliary problem, it is shown that the controls rendering the action functional stationary on arbitrary time horizons have a state feedback representation, via a verification theorem, that is consistent with the optimal control on short time horizons. An example is provided to illustrate application via a simple mass-spring system.