On the existence of equations of evolution (Q1061064)

The author considers nonlinear controlled systems (time-independent and non-autonomous) whose equation of evolution is not of the usually taken form \(u'=f(u,x)\) (u is the output, x the control and the prime denotes differentiation with respect to time) but is of the more general form (*) \(u'=f(u,x,x',...,x^{(n)})\). The need for an equation of this type occurs, for example, in the mechanics of inelastic continua. However, the existence of such equation cannot be assumed a priori for an arbitrary system; the value of n depends on the way in which the system smooth the control x. The author proves that, if the system is governed by a principle of determinism (i.e. the value of u at time \(t+\tau\) is determined by its value at time t and by history of the control in \([t,t+\tau])\), then it is described by an evolution equation of the type (*). The value n is such that the state u is continuously differentiable with respect to time whenever the control x is of class \(C^ n\).