Verifying Fundamental Solution Groups for Lossless Wave Equations via Stationary Action and Optimal Control
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Verifying Fundamental Solution Groups for Lossless Wave Equations via Stationary Action and Optimal Control Peter M. Dower1
· William M. McEneaney2
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A representation of a fundamental solution group for a class of wave equations is constructed by exploiting connections between stationary action and optimal control. By using a Yosida approximation of the associated generator, an approximation of the group of interest is represented for sufficiently short time horizons via an idempotent convolution kernel that describes all possible solutions of a corresponding short time horizon optimal control problem. It is shown that this representation of the approximate group can be extended to longer horizons via a concatenation of such short horizon optimal control problems, provided that the associated initial and terminal conditions employed in concatenating trajectories are determined via a stationarity rather than an optimality based condition. The construction is illustrated by its application to the approximate solution of a two point boundary value problem. Keywords Optimal control · Stationary action · Dynamic programming · Wave equations · Fundamental solution groups · Two point boundary value problems Mathematics Subject Classification 35L05 · 49J20 · 49L20
1 Introduction The action principle [1,13–15,17] is a variational principle underpinning modern physics that may be applied to a predefined notion of action to yield the equations
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Peter M. Dower [email protected] William M. McEneaney [email protected]
1
Department of Electrical & Electronic Engineering, University of Melbourne, Melbourne, VIC 3010, Australia
2
Department of Mechanical & Aerospace Engineering, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
123
Applied Mathematics & Optimization
of motion of a physical system and its underlying conservation laws. With a suitable definition of this action, the action principle specializes to Hamilton’s action principle, an important corollary of which states that any trajectory of an energy conserving system renders the corresponding action functional stationary in the calculus of variations sense. Consequently, Hamilton’s action principle can be interpreted as providing a characterization of all solutions of an energy conserving or lossless system. This interpretation motivates the development summarized in this work, with Hamilton’s action principle applied via an optimal control representation to construct the fundamental solution group corresponding to a lossless wave equation. The specific wave equation of interest is given by x¨ = −Λ x ,
(1)
where Λ is a linear, unbounded, positive, self-adjoint operator densely defined in an L 2 -space X , with a compact inverse Λ−1 ∈ L(X ). The results presented generalize . the recent work [9] from the specific Laplacian case Λ = −∂ 2 to any unbounded operator Λ satisfying the stated assumptions. In order to apply Hamilton’s action
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