Multi-Time Euler-Lagrange Dynamics∗

This paper introduces new types of Euler-Lagrange PDEs required by optimal control problems with performance criteria involving curvilinear or multiple integrals subject to evolutions of multidimensional-flow type. Particularly, the anti-trace multi-time Euler-Lagrange PDEs are strongly connected to the multi-time maximum principle. Section 1 comments the limitations of classical multi-variable variational calculus. Sections 2-3 refer to variational calculus with gradient variations and curvilinear or multiple integral functionals. Section 4 is dedicated to the study of the properties of multi-time Euler-Lagrange operator (affine changing of the Lagrangian, anti-trace multi-time Euler-Lagrange PDEs and new conservation laws). Section 5 formulates an application to multi-time rheonomic dynamics. Section 6 underlines the importance of the anti-trace multi-time Euler-Lagrange PDEs. Key–Words: gradient variations, multi-time Euler-Lagrange PDEs, multi-time maximum principle. Typing manuscripts, LATEX 1 Overview of classical multivariable variational calculus The foundations of variational calculus have been built using the classic Lagrange variation of an admissible function, but this determines some limitations that are not suitable for multi-time control theory. The most important limitation comes from the fact that the classical multi-variable variational calculus cannot be applied directly to create a multi-time maximum principle. In fact, the functionals given as multiple integrals, subject to general variation functions produce multi-variable Euler-Lagrange or Hamilton PDEs containing a trace (total divergence), which is not convenient for the conservation of the Hamiltonian. Indeed, the Hamiltonian is not a first integral for the multi-variable Hamilton PDEs, even in the autonomous case. Our multi-time control theory successfully overcomes the previous limitations [3]-[8]. This theory requires m-needle-shaped variations and complete integrability conditions as core issues. Adding new ideas in variational calculus via the gradient variations in curvilinear and multiple integrals and the anti-trace Euler-Lagrange or Hamilton PDEs, we have justified a multi-time maximum principle which is similar to the Pontryaguin maximum principle. The previous two types of functional variations ∗WSEAS Transactions on Mathematics, ., . (2007), ...-.... can be considered as ”celebrities” of the optimization theory, although the use of classical variations is hardly compatible with amplitude constraints, while m-needle-shaped variations are barely used in smooth optimization problems. 2 Curvilinear integral functional and gradient variations Let xi, i = 1, . . . , n denote the field variables on the target space Rn, let tα, α = 1, . . . ,m be the multitime variables on the source space Rm, and let xα = ∂xi ∂tα be the partial velocities. In this context, the jet bundle of order one is the manifold J1(Rm, Rn) = {(tα, xi, xα)}. Assume we are given a smooth completely integrable 1-form L = Lβ(x(t), xγ(t))dt , β, γ = 1, ...,m, t ∈ R + , called autonomous Lagrangian 1-form. The Lagrangian 1-form is determined by the Lagrange covector field Lβ(x(t), xγ(t)). The complete integrability conditions are ∂Lβ ∂xi ∂xi ∂tλ + ∂Lβ ∂xγ ∂xγ ∂tλ = ∂Lλ ∂xi ∂xi ∂tβ + ∂Lλ ∂xγ ∂xγ ∂tβ . Let Γ0,t0 be an arbitrary piecewise C 1 curve joining the points 0 and t0 in Rm + , and let Ω0,t0 be a parProceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 2007 66