In this paper, we consider a control system of the form (x) over dot = F(x)u, linear in the control variable u. Given a fixed starting point, we study a finite-horizon optimal control problem, where we want to minimize a weighted sum of an end-point cost and the squared 2-norm of the control. This functional induces a gradient flow on the Hilbert space of admissible controls, and we prove a convergence result by means of the Lojasiewicz-Simon inequality. Finally, we show that, if we let the weight of the end-point cost tend to infinity, the resulting family of functionals is Gamma-convergent, and it turns out that the limiting problem consists in joining the starting point and a minimizer of the end-point cost with a horizontal length-minimizer path.

A Gradient Flow Equation for Optimal Control Problems With End-point Cost / Scagliotti, A. - In: JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS. - ISSN 1079-2724. - (2022). [10.1007/s10883-022-09604-2]

### A Gradient Flow Equation for Optimal Control Problems With End-point Cost

#### Abstract

In this paper, we consider a control system of the form (x) over dot = F(x)u, linear in the control variable u. Given a fixed starting point, we study a finite-horizon optimal control problem, where we want to minimize a weighted sum of an end-point cost and the squared 2-norm of the control. This functional induces a gradient flow on the Hilbert space of admissible controls, and we prove a convergence result by means of the Lojasiewicz-Simon inequality. Finally, we show that, if we let the weight of the end-point cost tend to infinity, the resulting family of functionals is Gamma-convergent, and it turns out that the limiting problem consists in joining the starting point and a minimizer of the end-point cost with a horizontal length-minimizer path.
##### Scheda breve Scheda completa Scheda completa (DC)
2022
10.1007/s10883-022-09604-2
https://arxiv.org/abs/2107.00556
Scagliotti, A
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.11767/129530`
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