Virtual Lagrangian Construction Method for Systems of Partial Differential Equations

Abstract

This paper proposes a method of calculating a virtual Lagrangian for systems of partial differential equations that cannot be determined by standard Lagrangians. It is known that Lagrangians can be reconstructed from Euler-Lagrange equations by a homotopy operator. A homotopy operator is the dual operator of variational derivatives under a topological geometric assumption, called a vertically star-shaped. However, control systems are not always Euler-Lagrange equations, because of the dynamics of controllers and external forces. First, we introduce the concept of jet bundles to treat variational problems in a uniform way. Then we show that systems on vertically star-shaped regions with zero boundary conditions can be decomposed into two systems. One of them is a subsystem determined by virtual Lagrangians, which we call an exact subsystem, and the other is never introduced from calculus of variations, which we call a dual-exact subsystem. Next, we clarify the control input that turns given systems into exact systems, which we call an exactize control.