Generalized solutions of nonlinear optimization problems with impulse controls. II: Representation of solutions by differential equations with measure (Q1285400)

The nonlinear dynamic system considered is described by the differential equation \[ \dot x(t)= F(x(t), u(t), w(t),t),\quad x(0)= x_0, \] where \(F(\cdot,\cdot,\cdot,\cdot)\) is a given function, \(x(t)\in \mathbb{R}^n\) is the state variable at time \(t\in [0,T]\) and \(u(\cdot)\), \(w(\cdot)\) are measurable control functions such that \(u(t)\in U\), \(U\) is a bounded and closed subset of \(R^k\), \(w(t)\in W\), \(W\) is not bounded in norm of \(\mathbb{R}^m\) and \(\int^T_0\|w(t)\|dt< M\) for some given constant \(M\). In the earlier paper [\textit{B. M. Miller}, Autom. Remote Control 56, No. 4, Pt. 1, 505-516 (1995); translation from Avtom. Telemekh. 1995, No. 4, 62-76 (1995; Zbl 0928.93022)] a concept of a generalized solution of the equation is given and, under additional assumptions, a solution is obtained by a discontinuous time substitution method. Based on this result, in the present paper, a nonlinear equation with measure is derived. The equation describes arbitrary generalized solutions.