Numerical study of polynomial feedback laws for a bilinear control problem

In this paper, we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term in a bidimensional domain. The existence, uniqueness and regularity of strong solutions of this model are deduced, proving the existence of a global optimal solution. Afterwards, we derive first-order optimality conditions by using a Lagrange multipliers theorem.

In this work, a new method for solving an optimal semi-active control problem is formulated. It extends previously published results by addressing a continuous-time bilinear quadratic regulator control problem, defined for a structure that is controlled by multiple controlled hydraulic dampers and subjected to external, a priori known deterministic excitation input. The dynamic system is modeled by a linear state-space model, accompanied by constraints on the control forces that reflect the semi-active nature of the prescribed dampers. Next, the control force is written in an equivalent bilinear form, thereby leading to a bilinear state-space model. The optimal control problem is defined by this dynamic bilinear representation, control force constraints, and a performance index that is quadratic in the states and the equivalent damping gains. Namely, as each device has only two allowable operation modes, its control signal’s range is a two-object set. Krotov’s method is used for solving this problem. This method requires to formulate a function’s sequence with special properties and a suitable minimizing feedback. By doing so, a process improvement becomes possible and an improving sequence can be formulated. This sequence of improving functions and the suitable minimizing feedback, which enable the use of Krotov’s method in this case, are formulated in the context of the addressed problem. The results are encapsulated in a convergent algorithm whose outcome is an approximation of a candidate optimal feedback. Numerical example is given to demonstrate the use of the suggested results for a CBQR control design of a structure, subjected to seismic input.

An optimal control problem for the stationary Navier–Stokes equations with variable density is studied. A bilinear control is applied on the flow domain, while Dirichlet and Navier boundary conditions for the velocity are assumed on the boundary. As a first step, we enunciate a result on the existence of weak solutions of the dynamical equation; this is done by firstly expressing the fluid density in terms of the stream-function. Then, the bilinear optimal control problem is analyzed, and the existence of optimal solutions are proved; their corresponding characterization regarding the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.

<abstract> This paper discusses some a priori error estimates of bilinear elliptic optimal control problems based on the finite volume element approximation. A case-based numerical example serves to discuss with optimal $ L^2 $-norm error estimates and $ L^{\infty} $-norm error estimates, and supports two key insights. First, the approximate orders for the state, costate and control variables are $ O(h^2) $ in the sense of $ L^{2} $-norm. Second, the approximate orders for the state, costate and control variables are $ O(h^2\sqrt{|lnh|}) $ in the sense of $ L^{\infty} $-norm. </abstract>

Semi-explicit solution to a partially observed stochastic bilinear system control problem is given. The system is bilinear in respect to the control inputs, and corrupted by additive Wiener processes in the state and measurements equations. Beside the unobserved state the initial state is an unknown parameter. The optimal controls in a mean square sense are derived in dependence on the optimal value function that is solved numerically from the Bellman equation and saved in a look-up table for prompt calculation of optimal controls afterwards. The control effect is demonstrated in a simulation experiment where the controls remove the initial uncertainty and stabilise the process at the target level effectively.

In this paper, we investigate the saddle-point equilibrium for continuous singular bilinear system control problem in the infinite time, conditions for the existence of saddle-point equilibrium strategies are given, and the saddle-point equilibrium strategies are derived. In the end, the simulation of a numerical example is presented.

An alternative formulation is presented for a class of partially observed bilinear stochastic control problems which is described by three sets of stochastic differential equations: one for the system to be controlled, one for the observer, and one for the control process which is driven by the observation process. With this formulation, the stochastic control problem is converted to an equivalent deterministic identification problem of control gain matrices. Using standard variation arguments, the necessary conditions of optimality on the basis of which the optimal control parameters can be determined are obtained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Computation of nonlinear balanced realization and model reduction based on Taylor series expansion

Solar energy constitutes one of the most abundant energy resources. Advanced control techniques can have an important role in the promotion of such renewable energies by improving their efficiency. In this paper, a finite-horizon optimal controller is proposed to solve the problem of tracking a prescribed temperature profile in a solar collector parabolic trough (SC). The SC is represented by a first-order hyperbolic PDE, where the control input is the pump flow rate which generates a bilinear control problem. Based on the control parameterization approach, an approximate optimal controller is designed in the infinite-dimensional framework. The gradients of the cost function are formulated analytically, based on the sensitivity method. The approximate nonlinear optimization problem is then solved numerically using sequential quadratic programming. Simulation results illustrate the effectiveness of the designed control technique.

Optimal control of convection-cooling and numerical implementation

In this paper, we consider a class of optimal control problems in which the system is to be determined in an optimal way. This class of problems involves the choice of a fixed number of switching time points which divide the system’s time horizon into a number of time periods. For each of these time periods, a subsystem is selected, from a finite number of given candidate subsystems, to run during that time periods. The choice of the switching points and the selection of the subsystems are carried out in such a way that a given cost functional is minimized. We consider only problems involving ordinary differential equations over a finite time horizon. A computational method is developed for solving these problems. In our method, the candidate subsystems are combined into a single system by considering their ‘linear combination’. By introducing suitable constraints on the coefficients of the linear combination and using a time rescaling technique, the original problem is transformed into an equivalent optimal control problem with system parameters. An algorithm is proposed for solving this transformed problem and the required gradient formulae are derived. To show the effectiveness of the method, a numerical example is solved.

This paper deals with discrete-time Markov decision processes (MDPs) under constraints where all the objectives have the same form of expected total cost over the infinite time horizon. The existence of an optimal control policy is discussed by using the convex analytic approach. We work under the assumptions that the state and action spaces are general Borel spaces, and that the model is nonnegative, semicontinuous, and there exists an admissible solution with finite cost for the associated linear program. It is worth noting that, in contrast to the classical results in the literature, our hypotheses do not require the MDP to be transient or absorbing. Our first result ensures the existence of an optimal solution to the linear program given by an occupation measure of the process generated by a randomized stationary policy. Moreover, it is shown that this randomized stationary policy provides an optimal solution to this Markov control problem. As a consequence, these results imply that the set of randomized stationary policies is a sufficient set for this optimal control problem. Finally, our last main result states that all optimal solutions of the linear program coincide on a special set with an optimal occupation measure generated by a randomized stationary policy. Several examples are presented to illustrate some theoretical issues and the possible applications of the results developed in the paper.

The optimal control of a class of stochastic parabolic systems is studied. This class includes systems with noise depending on spatial derivatives of the state, Neumann boundary control, and Dirichlet boundary observation, and extends a class of stochastic systems with distributed control studied by Da Prato [3] and Da Prato and Ichikawa [4]. The work is based on the direct study of the Riccati equation arising in the optimal control problem over finite time horizon. The problem over infinite time horizon and the corresponding algebraic Riccati equation are also considered.

In this article we apply the optimal and the robust control theory to the sine-Gordon equation. In our case the control is given by the boundary conditions and we work in a finite time horizon. We present at the beginning the optimal control problem and we derive a necessary condition of optimality and we continue by formulating a robust control problem for which existence and uniqueness of solutions are derived.

Existing methods for nonlinear robust control often use scenario‐based approaches to formulate the control problem as large nonlinear optimization problems. The optimization problems are challenging to solve due to their size, especially if the control problems include time‐varying uncertainty. This paper draws from local reduction methods used in semi‐infinite optimization to solve robust optimal control problems with parametric and time‐varying uncertainty. By iteratively adding interim worst‐case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. We show that the local reduction method for optimal control problems consists of solving a series of simplified optimal control problems to find worst‐case constraint violations. In particular, we present examples where local reduction methods find worst‐case scenarios that are not on the boundary of the uncertainty set. We also provide bounds on the error if local solvers are used. The proposed approach is illustrated with two case studies with parametric and additive time‐varying uncertainty. In the first case study, the number of scenarios obtained from local reduction is 101, smaller than in the case when all extreme scenarios are considered. In the second case study, the number of scenarios obtained from the local reduction is two compared to 512 extreme scenarios. Our approach was able to satisfy the constraints both for parametric uncertainty and time‐varying disturbances, whereas approaches from literature either violated the constraints or became computationally expensive.