A fundamental solution for an infinite dimensional two-point boundary value problem via the principle of stationary action

A new approach for solving two-point boundary value problems for conservative infinite dimensional systems is investigated. This new approach seeks to exploit the principle of stationary action in reformulating and solving such problems in the framework of optimal control. A specific and highly simplified problem involving a one dimensional wave equation is considered. The construction of a solution to an attendant optimal control problem is identified as crucial in applying this new approach.

[1]  R. Feynman,et al.  Space-Time Approach to Non-Relativistic Quantum Mechanics , 1948 .

[2]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[3]  William M. McEneaney,et al.  A max-plus method for optimal control of a diffusion equation , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[4]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[5]  Edwin F. Taylor,et al.  When action is not least , 2007 .

[6]  William M. McEneaney,et al.  The Principle of Least Action and Solution of Two-Point Boundary Value Problems on a Limited Time Horizon , 2013, SIAM Conf. on Control and its Applications.

[7]  William M. McEneaney,et al.  Max-plus methods for nonlinear control and estimation , 2005 .

[8]  S. Bernau.  The square root of a positive self-adjoint operator , 1968, Journal of the Australian Mathematical Society.

[9]  William M. McEneaney,et al.  The Principle of Least Action and Fundamental Solutions of Mass-Spring and N-Body Two-Point Boundary Value Problems , 2015, SIAM J. Control. Optim..

[10]  Huan Zhang,et al.  A max-plus method for the approximate solution of discrete time linear regulator problems with non-quadratic terminal payoff , 2013, SIAM Conf. on Control and its Applications.

[11]  William M. McEneaney,et al.  A new fundamental solution for differential Riccati equations arising in control , 2008, Autom..

[12]  William M. McEneaney,et al.  A max-plus based fundamental solution for a class of infinite dimensional Riccati equations , 2011, IEEE Conference on Decision and Control and European Control Conference.