CONSTRAINED LINEAR-QUADRATIC CONTROL PROBLEMS OVER TIME SCALES AND WEAK NORMALITY

Time scale linear-quadratic control problems with affine mixed state-control and joint endpoints equality constraints are considered. Without any controllability assumption, it is shown that the feasible pairs at which the first variation vanishes are exactly the feasible pairs that satisfy the weak maximum principle (called "extremals") with $\lambda_0 = 1$. In this case, we say the problem is "weakly normal" at such feasible pairs. When a certain matrix function $S(\cdot)$ has invertible images, the weak-normality condition at $(\bar x, \bar u)$ with associated adjoint variable $\bar p$ is also equivalent to $(\bar x, \bar p)$ solving the corresponding non-homogeneous symplectic boundary value problem and to $\bar u$ being a certain affine combination of this solution. In this equivalence, the invertibility of the corresponding matrices $S(t)$ is not needed when the linear-quadratic problem is itself symplectic. As an application, it is established that without any controllability assumption, the optimality in linear-quadratic problems is characterized in terms of either the weak normality condition, or the solvability of the corresponding symplectic boundary value. These results are obtained for linearquadratic control problems with or without shift in the state variable and are new not only for the time scale setting but also for the continuous time and discrete time settings.