Sufficient optimality condition for linear control of plants described by differential equations with a measure (Q1058128)

This paper considers sampled-data control of plants whose dynamics can be described by differential equations with a measure as follows: \[ dx(t)=A(t)x(t)dt+B(t)du(t),\quad x(0-)=x_ 0, \] where \(x(t)\in R^ n\), \(A(.)\in L^ 2(0,T;R^ n\times R^ n),\) \(B(.)\in C([0,T];R^ n\times R^ m)\) and \(u(t)\in R^ m\) satisfies \(v(t)=_{[0,T]}u(s)<+\infty\) and \(u(t+)=u(t)\). The set of allowed controls \[ \Omega =\{u:_{[0,T]}u(s)<+\infty,\quad u(t)-u(s-)\in K,\quad t\geq s\}, \] where K is a convex cone in \(R^ n\) and the cost functional is \(J[x(.),u(.)]=\phi_ 0(x(T),x(0-),v(T))\) where \(\phi_ 0\) is convex in all the variables. In addition, there are boundary conditions \(S_ k(x(T),x(0-),v(T))\leq 0,\) \((k=1,...,l)\), where the \(S_ k's\) are also convex in all the variables. A necessary optimality condition of maximum principle type is obtained, and under moderate constraints it is also a sufficient condition. Results can be applied to the problem of control of discrete observations that arises when dynamic systems must be controlled that operate under disturbances and with incomplete information about the system state.