Controlling discrete-continuous models of catastrophes (Q1385849)

The article under review may be best regarded as a general outline of better things to come. A variational approach to controlled processes which may become catastrophic uses representation in the form of discrete scenes, in which the state of the system is given as a function of the initial conditions and of the control variables. Here, the author assumes continuity and twice differentiability of the transformation functions mapping the \(i\)-th scene into the \((i+1)\)-th scene. The adjoint variables are defined by means of the Hamiltonian of the process, and the equations of motion are exactly Hamilton's canonical equations. The functional (possibly cost functional) of the process \(J\) is a given functional of the phase coordinates related to the last scene (call it instant) of the catastrophic process. The variation of the functional \(J\) is the virtual work plus the dual work. (For example in the simplest thermodynamic process this would represent the terms \(p\delta v+v \delta p+\dots\;.\)) Transition of the scene, transition of the boundary plus the Hamilton canonical equations and positivity of such virtual work terms constitute necessary conditions leading to a minimum condition for minimization of the process functional, and suggest a new form of maximality (or minimiality) principle of Pontryagin type for the corresponding optimal controls. Hopefully this article will be followed by specified examples of application and/or more detailed explanations in future publications.