Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property

Abstract

We consider a vibrating string that is fixed at one end with Neumann control action at the other end. We investigate the optimal control problem of steering this system from given initial data to rest, in time T , by minimizing an objective functional that is the convex sum of the L 2 -norm of the control and of a boundary Neumann tracking term.We provide an explicit solution of this optimal control problem, showing that if the weight of the tracking term is positive, then the optimal control action is concentrated at the beginning and at the end of the time interval, and in-between it decays exponentially. We show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term. This implies that, if the time T is large the optimal trajectory approximately consists of three arcs, where the first and the third short-time arcs are transient arcs, and in the middle arc the optimal control and the corresponding state are exponentially close to 0. This is an example for a turnpike phenomenon for a problem of optimal boundary control. If T = +∞ (infinite horizon time problem), then only the exponentially decaying component of the control remains, and the norms of the optimal control action and of the optimal state decay exponentially in time. In contrast to this situation if the weight of the tracking term is zero and only the control cost is minimized, then the optimal control is distributed uniformly along the whole interval [0, T ] and coincides with the control given by the Hilbert Uniqueness Method.

Minimal controllability time for the heat equation under unilateral state or control constraints

“…According to Remark 11 and as a consequence of the comparison principle for the solutions of the heat equation, the minimal time for the control of the heat equation with state constraints coincides with the minimal time arising when the constraints are imposed only on the control. According to Lemma 2 in Section 2.6, in both situations, this minimal time satisfies lower estimates given by (10). Moreover, for the Dirichlet control problem we can take u0 = u1, and for the Neumann control problem we can take v0 = −v1.…”

“…We have recently extended this analysis in [22] to infinite dimension, involving in particular the case of heat equations (see also [10] for a specific analysis on the wave equation). Here, however, the context is slightly different because we minimize the time, and therefore the final time T is not expected to be large as in the above-mentioned references.…”

Large time control and turnpike properties for wave equations

“…In the last decades there has been a growing interest and a significant number of papers was published in this area, both for time-discrete and time-continuous finite dimensional systems (see, for instance, Anderson and Kokotovic, 1987;Artstein and Leizarowitz, 1985;Carlson, Haurie, and Jabrane, 1987;Carlson, Haurie, and Leizarowitz, 1991;Dorfman et al, 1958;Faulwasser, Korda, Jones and Bonvin, 2015;Grüne, 2013;Grüne and Müller, 2016;Gugat, Trélat, and Zuazua, 2016;McKenzie, 1963;Porretta and Zuazua, 2013;Rapaport and Cartigny, 2004;Trélat and Zuazua, 2015;Zaslavski, 2006;2014;2015; Lou and Wang and references therein).…”

“…In particular, in Porretta and Zuazua (2013) , the wave equation was shown to fulfil the exponential turnpike property provided the control satisfies the so-called Geometric Control Condition (GCC) (see Zuazua, 2006 ), ensuring that all rays of Geometric Optics enter the control subdomain in an uniform time. The particular case of the linear 1 − d wave equation was treated in more detail in Gugat et al (2016) .…”

Model predictive control strategy of energy-water management in urban households

“…This problem is called the turnpike phenomenon [58]. Turnpike property has been described by Faulwasser et al [59] as the phenomenon where the optimal solution in many finite-horizon optimal controlproblems for different initial conditions approach the purlieus of the best steady state but might leave it towards the end of the control horizon.…”