Abstract
The existence theorems of this Chapter concern control systems as well as problems of the calculus of variations which are linear in the state variables, but not necessarily linear in the controls and no convexity assumptions are required. Theorems of this type were first noted by L. W. Neustadt, and they are based on set theoretical considerations due to A. A. Lyapunov. We first prove in Section 16.1 some theorems of the Lyapunov type, and we use them in Section 16.2 to prove Neustadt type existence theorems for the bounded case. In this situation we prove in Section 16.3 that there always are bang-bang solutions. In Sections 16.5–6 we handle the unbounded case, and in Section 16.7 problems of the calculus of variations.
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