Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control

Abstract

We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether's theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, and which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples given.

The Second Euler-Lagrange Equation of Variational Calculus on Time Scales

“…Standard Noetherian constants of motion are violated due to the presence of a new term that depends on the graininess µ(t) of the time scale, while in the classical context µ(t) ≡ 0. The importance of Noether's conservation laws in the calculus of variations, optimal control theory, and its applications in engineering, are well recognized [12], [21], [22], [33]. Their role on the general context of optimal control on time scales is an entirely open area of research.…”

The Hahn Quantum Variational Calculus

“…It is worth to mention that due to the constraints on the values of the controls (u 1 (t), u 2 (t) ∈ Ω = [−1, 1]), a theory based on necessary optimality conditions to solve problem (31)- (33) does not exist at the moment. We begin noticing that problem (31)-(33) is variationally invariant according to [45] under the one-parameter family of transformations…”

Computing ODE symmetries as abnormal variational symmetries

“…Here we remark that the abnormal variational symmetries (i.e. the ones associated with ψ 0 = 0) obtained by the method introduced in [5] provide symmetries for ordinary differential equations.…”