On Sparse Optimal Control for General Linear Systems

In this paper, we investigate an <inline-formula><tex-math notation="LaTeX">$L^0$</tex-math></inline-formula> optimization problem with constraints in a form of Volterra integral equation and the <inline-formula><tex-math notation="LaTeX">$L^{\infty }$</tex-math></inline-formula> norm. In particular, the equivalence theorem among the <inline-formula><tex-math notation="LaTeX">$L^p$</tex-math></inline-formula> optimizations with <inline-formula><tex-math notation="LaTeX">$p\in [0, 1]$</tex-math></inline-formula> is derived, which provides the following twofold extension of the existing results: First, these theoretical results enable us to solve sparse optimal control problems without imposing the finite dimensionality of the system to be controlled, which was the crucial assumption for the derivation of the existing results. Second, the relationship between the partial state constrained problem and output controllability is newly characterized.

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