Abstract
Let \( \bar{u}(t) \)be a control that satisfies the infinite-dimensional version of Pontryagin’s maximum principle for a linear control system, and let \( {z}(t) \)be the costate associated with \( \bar{u}(t) \). It is known that integrability of \( {z}(t) \)in the control interval [0, T] guarantees that \( \bar{u}(t) \)is time and norm optimal. However, there are examples where optimality holds (or does not hold) when \( {z}(t) \)is not integrable. This paper presents examples of both cases for a particular semigroup (the right translation semigroup in \( {L^2}(0,\infty )\)).
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Mathematics Subject Classification (2000). 93E20, 93E25.
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Fattorini, H.O. (2011). Time and Norm Optimality of Weakly Singular Controls. In: Escher, J., et al. Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 80. Springer, Basel. https://doi.org/10.1007/978-3-0348-0075-4_12
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DOI: https://doi.org/10.1007/978-3-0348-0075-4_12
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