Cheap Control, Singular Arcs, and Singular Perturbations

Abstract

Consider the control problem consisting of the state equation

$$\dot x\, =\,Ax\,+\,Bu\,,\quad 0\,\leqslant\,t\leqslant 1,$$

((1))

, with the initial state x(0) prescribed, and with the scalar cost

$$J(\varepsilon)\,=\frac{1}{2}\int_0^1{\left[{{x^T}(t)Qx(t)\,+\,{\varepsilon ^2}{u^T}(t)Ru(t)}\right]dt}$$

((2))

to be minimized. Here, x and u are vectors of dimension n and r, respectively, Q and R are symmetric matrices, Q is positive semidefinite, R is positive definite, and e is a small positive parameter. Since ε² multiplies the control part of the cost, control is cheap relative to state (cf. Lions (1973) which discusses cheap control for analogous problems in partial differential equations).