Optimal control of causal differential-algebraic systems
The paper deals with optimal control of systems whose dynamics are of the type \(\dot{x} = f (t,x,y,u)\), \(g(t,x,y,u)=0\), where \(u\) is the control, \(x\) and \(y\) are, respectively, the ``slow'' and the ``fast'' components of the state. One system of this type has index \(k\) if one needs to differentiate the algebraic constraint, \(g(t,x,y,u)=0\), \((k-1)\) times in order to obtain the underlying system of ODE's. The authors present regularity conditions that allow the given optimal control problem to be reduced to an equivalent problem with dynamics of the type \(\dot{x} = f (t,x,u)\) by using an implicit function argument. This reduction is used to obtain sufficient conditions for the existence of optimal controls and necessary optimality conditions under the form of suitable versions of the maximum principle. This is done separately for systems of index 1, 2 and 3. Beyond index 3 the method becomes impractical for general cases because of the complexity of the conditions involved. An application to an index-3 problem arising in robotics is presented.
scientific article; zbMATH DE number 1797501
scientific article