Greedy optimal control for elliptic problems and its application to turnpike problems

This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-018-1005-zWe adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter-space to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. The turnpike property (which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough) allows using the elliptic greedy choice of the parameters in the parabolic setting too. We present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithmsThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694126-DyCon). Part of this research was done while the second author visited DeustoTech and Univesity of Deusto with the support of the DyCon project. The second author was also partially supported by Croatian Science Foundation under ConDyS Project, IP-2016-06-2468. The work of the third author was partially supported by the Grants MTM2014-52347, MTM2017-92996 of MINECO (Spain) and ICON of the French AN

Stochastic optimization methods for the simultaneous control of parameter-dependent systems

We address the application of stochastic optimization methods for the simultaneous control of parameter-dependent systems. In particular, we focus on the classical Stochastic Gradient Descent (SGD) approach of Robbins and Monro, and on the recently developed Continuous Stochastic Gradient (CSG) algorithm. We consider the problem of computing simultaneous controls through the minimization of a cost functional defined as the superposition of individual costs for each realization of the system. We compare the performances of these stochastic approaches, in terms of their computational complexity, with those of the more classical Gradient Descent (GD) and Conjugate Gradient (CG) algorithms, and we discuss the advantages and disadvantages of each methodology. In agreement with well-established results in the machine learning context, we show how the SGD and CSG algorithms can significantly reduce the computational burden when treating control problems depending on a large amount of parameters. This is corroborated by numerical experiments

Exponential Turnpike property for fractional parabolic equations with non-zero exterior data

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Control problems play a key role in engineering, economics and sciences. To be more precise, in climate sciences, often times, relevant problems are formulated in long time scales, so that, the problem of possible asymptotic behaviors when the time-horizon goes to infinity becomes natural. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question arises: Do time averages of optimal controls and trajectories converge to the stationary optimal controls and states as the time-horizon goes to infinity? This question is very closely related to the so-called turnpike property that shows that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In the present paper we deal with heat equations with non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the fractional Laplace operator $(-\Delta)^s$ ($0<s<1$). We prove the turnpike property for the nonlocal Robin optimal control problem and the exponential turnpike property for both Dirichlet and nonlocal Robin optimal control problems

The turnpike property in semilinear control

An exponential turnpike property for a semilinear control problem is proved. The state-target is assumed to be small, whereas the initial datum can be arbitrary. Turnpike results are also obtained for large targets, requiring that the control acts everywhere. In this case, we prove the convergence of the infimum of the averaged time-evolution functional towards the steady one. Numerical simulations are performed

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

Challenges in Optimization with Complex PDE-Systems (hybrid meeting)

The workshop concentrated on various aspects of optimization problems with systems of nonlinear partial differential equations (PDEs) or variational inequalities (VIs) as constraints. In particular, discussions around several keynote presentations in the areas of optimal control of nonlinear or non-smooth systems, optimization problems with functional and discrete or switching variables leading to mixed integer nonlinear PDE constrained optimization, shape and topology optimization, feedback control and stabilization, multi-criteria problems and multiple optimization problems with equilibrium constraints as well as versions of these problems under uncertainty or stochastic influences, and the respectively associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, aspects of optimal control of data-driven PDE constraints (e.g. related to machine learning) were addressed