Turnpike Property for Generalized Linear-Quadratic Optimal Control Problem

The turnpike phenomenon describes the long time behavior of optimally controlled systems whose optimal trajectories over a sufficiently large time horizon stay for most of the time close to a prescribed trajectory of the system. This thesis is devoted to the characterization of the turnpike property for generalized LQ optimal control problem. Through our research, we derive both sufficient and necessary conditions for the turnpike property in infinite dimensional setting. It is shown that the turnpike property is closely related to certain structural properties of the control system. In particular, we deduce an equivalent condition of the turnpike property in terms of the exponential stabilizability and detectability of the system for finite dimensional case and point spectrum case. We also show in our thesis that the turnpike property for generalized LQ optimal control problem is equivalent to the turnpike property for LQ optimal control problem plus an algebraic condition. Next, we investigate the applications of our results to the generalized LQ optimal control problem subject to the parabolic equations, wave equations, delay equations and in relation with model predictive control schemes

The turnpike property in finite-dimensional nonlinear optimal control

Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry. They refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal control problem. We provide in this paper a general version of a turnpike theorem, valuable for nonlinear dynamics without any specific assumption, and for very general terminal conditions. Not only the optimal trajectory is shown to remain exponentially close to a steady-state, but also the corresponding adjoint vector of the Pontryagin maximum principle. The exponential closedness is quantified with the use of appropriate normal forms of Riccati equations. We show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method, or a shooting method. In particular, we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector, not at the initial time, but at the middle of the trajectory

Integral and measure-turnpike properties for infinite-dimensional optimal control systems

We first derive a general integral-turnpike property around a set for infinite-dimensional non-autonomous optimal control problems with any possible terminal state constraints, under some appropriate assumptions. Roughly speaking, the integral-turnpike property means that the time average of the distance from any optimal trajectory to the turnpike set con- verges to zero, as the time horizon tends to infinity. Then, we establish the measure-turnpike property for strictly dissipative optimal control systems, with state and control constraints. The measure-turnpike property, which is slightly stronger than the integral-turnpike property, means that any optimal (state and control) solution remains essentially, along the time frame, close to an optimal solution of an associated static optimal control problem, except along a subset of times that is of small relative Lebesgue measure as the time horizon is large. Next, we prove that strict strong duality, which is a classical notion in optimization, implies strict dissipativity, and measure-turnpike. Finally, we conclude the paper with several comments and open problems

Steady-state and periodic exponential turnpike property for optimal control problems in hilbert spaces

First Published in SIAM Journal on Control and Optimization in Volume 56, Issue 2, 2018, Pages 1222-1252, published by the Society for Industrial and Applied Mathematics (SIAM)In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reects the fact that, in large control time horizons, the optimal state and control and adjoint state remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking termsThe authors acknowledge the nancial support by the grant FA9550-14-1-0214 of the EOARD-AFOSR. The second author was partially supported by the National Natural Science Foundation of China under grants 11501424 and 11371285. The third author was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, FA9550-15-1-0027 of AFOSR, the MTM2014-52347 and MTM2017-92996 grants of the MINECO (Spain), and ICON of the French AN

Long time control with applications

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 24-04-2020This thesis is concerned with the study of some control problems in a large time horizon. The first part of the thesis is devoted to controllability of Partial Differential Equations under state and/or control constraints. In chapter 4, we address the controllability under positivity constraints of semilinear heat equations. We firstly obtain steady state controllability, by employing a ``stair-case argument''. Then, supposing dissipativity of the free dynamics, we extend our previous result to constrained controllability to trajectories. In any case, the targets must be defined by positive controls. We prove further the positivity of the minimal controllability time under positivity constraints, by applying a new method, based on the choice of a particular test function in the definition of weak solutions to evolution equations. Hence, despite the infinite velocity of propagation for parabolic equations, a waiting time phenomenon occurs in the constrained case. In chapter 5, controllability under positivity constraints is analyzed for wave equations. In this case, the zero state is reachable, by nonnegative controls. In chapter 6, we get a global turnpike result for an optimal control problem, governed by a semilinear heat equation. The running target in the cost functional is required to be small, whereas the initial datum for the evolution equation can be chosen arbitrarily. This is done by combining the available local results [116, 137], with an estimate of the L1 norm of the optima (uniform in the time horizon) and an estimate of the time needed to get close to the turnpike. If the target is large, we produce an example, where the steady problem admits (at least) two solutions (chapter 7). In chapter 8, we present an application of stabilization/turnpike theory to a problem of rotor balancingEsta tesis concierne el estudio de algunos problemas de control en un largo horizonte temporal. La primera parte de la tesis está dedicada a la controlabilidad de Ecuaciones en Derivadas Parciales bajo restricciones de estado y/o control. En el capítulo 4, abordamos la controlabilidad bajo restricciones de positividad para la ecuación del calor semilineal. En primer lugar, obtenemos la controlabilidad entre estados estacionarios, mediante el uso de un ``stair-case argument''. Luego, suponiendo disipatividad en la dinámica libre, extendemos nuestro resultado anterior a la controlabilidad bajo restricciones hacia trayectorias. En cualquier caso, los targets deben definirse mediante controles positivos. Ademas, probamos la positividad del tiempo mínimo de controlabilidad bajo restricciones de positividad, mediante la aplicación de un nuevo método, basado en la elección de una función test particular en la definición de solucione débil para la ecuación de evolución. Por lo tanto, a pesar de la velocidad infinita de propagación para las ecuaciones parabólicas, se produce un fenómeno de tiempo de espera en el caso restringido. En el capítulo 5, la controlabilidad bajo restricciones de positividad se analiza para la ecuación de ondas. En este caso, el estado cero es alcanzable por controles positivos. En el capítulo 6, obtenemos un resultado de turnpike global para un problema de control optimo, sujeto a una ecuación del calor semilineal. En este caso, requerimos que el target en el funcional de coste sea pequeño, mientras que el dato inicial para la ecuación de evolución se puede elegir arbitrariamente. Esto se realiza combinando los resultados locales disponibles en [116, 137], con una estimación de la norma L1 para los optimos (uniforme en el horizonte temporal) y una estimación del tiempo necesario para acercarse al turnpike. Para el caso de target grande, damos un ejemplo, donde el problema estacionario admite (al menos) dos soluciones (capítulo 7). En el capítulo 8, presentamos una aplicación de la teoría de estabilización/turnpike a un problema de equilibrio para un rotorThis thesis has been mainly funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 694126-DyCon), and for an applied research secondment by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 77782