Singular Optimal Control Problems Research Articles

Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.

Optimization Modeling for Advanced Syngas to Olefin Reactive Systems

Reactor designs with mixed catalysts play an important role in transforming a multiple reactor system to single-shot reactors. In addition to savings in capital and ease of implementation, single-shot reactors are useful to break equilibrium limitations, thereby increasing the yield and selectivity of desired product as shown in previous studies. However, the nonlinear and highly exothermic nature of mixed-catalyst systems makes it difficult for commercial process simulation and optimization tools to optimize these systems. This study describes the development and application of optimization strategies for mixed-catalyst, single-shot reactors for syngas to olefin (STO) processes. Finding the optimal catalyst distribution is challenging and requires advanced solution strategies for singular optimal control problems, which are poorly conditioned and often lead to flat response surfaces. The graded bed and partial-moving finite-element approaches are used to find the optimal catalyst distribution that maximizes the olefins yield. A 1.3% increase in the yield is observed from one zone to three zones. The yield further improves from three zones to the exact infinite dimensional solution by 0.2%. This improvement can be realized in practice by changing only the catalyst distribution, without any extra investment. Finally, the results suggest that a mixed-catalyst single shot reactor bed can be applied to other reaction mechanisms to increase reactor performance.

A switched system approach for the direct solution of singular optimal control problems

Optimal control problems where the control variable appears linearly in the formulation lead to singular control arcs that introduce analytical and numerical complications during the solution process. In this work, the switching behavior (switching between singular and nonsingular control arcs) is exploited to solve singular optimal control problems using a direct method. The problem is reformulated as a switched system optimal control problem where each control arc defines a different subsystem, i.e., lower and upper limits and a singular arc. The switching between control arcs is handled using binary functions that depend on time. The switched system is embedded into a larger family by relaxing the binary functions to avoid integer variables. The variable switching times at which the control switches between arcs are mapped to fixed points in a new time domain using a time-scaling transformation. A direct transcription method along with control parametrization is applied to transform the embedded problem into a nonlinear programming formulation that contains few variables and constraints. A penalty term is introduced to obtain binary solutions for the optimal selection of control arcs. The proposed reformulation avoids the use of mesh refinement techniques and continuation methods, and allows to solve challenging problems in very short CPU times with high accuracy.

Auxiliary Minimum Covariance Resolution of Infinite-Order Singular Arcs

Singular arcs are a common feature in optimal control problems when the standard necessary conditions fail to reveal a candidate control. In some cases, a singular arc can be proven to be nonoptimal and discarded. In other cases, the solution can be resolved through additional convexity conditions on the Hamiltonian. A less common, but nonetheless interesting, scenario occurs when the singular arc appears with infinite order. This situation becomes apparent when the problem definition lacks sufficient information to identify a unique solution. In this work, we consider conditions under which singular optimal control problems may exhibit indeterminate behavior due to insufficiency of the cost objective and boundary conditions. The resulting problem is shown to be singular, but in a degenerate sense wherein the necessary conditions yield no new information. A relaxation approach is developed to repose the problem with augmented covariance dynamics. The method permits finite-order arcs producing a well-posed problem. The approach has relevance to engineering systems, which are almost always subject to some level of uncertainty in initial states or parameters. Examples are provided based on classical problems in optimal control, as well as aerospace mission design.

Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply

We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. Further, we derive a input-state turnpike towards a subspace for optimal control of port-Hamiltonian ordinary differential equations with a feed-through term and a turnpike property for the corresponding adjoint states towards zero. In an appendix we characterize the class of dissipative Hamiltonian matrices and pencils.

Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation

A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre-Gauss-Radau (LGR) collocation. The method is divided into several parts. First, a structure detection method is developed that identifies switch times in the control and analyzes the corresponding switching function for segments where the solution is either bang-bang or singular. Second, after the structure has been detected, the domain is decomposed into multiple domains such that the multiple-domain formulation includes additional decision variables that represent the switch times in the optimal control. In domains classified as bang-bang, the control is set to either its upper or lower limit. In domains identified as singular, the objective function is augmented with a regularization term to avoid the singular arc. An iterative procedure is then developed for singular domains to obtain a control that lies in close proximity to the singular control. The method is demonstrated on four examples, three of which have either a bang-bang and/or singular optimal control while the fourth has a smooth and nonsingular optimal control. The results demonstrate that the method of this paper provides accurate solutions to problems whose solutions are either bang-bang or singular when compared against previously developed mesh refinement methods that are not tailored for solving nonsmooth and/or singular optimal control problems, and produces results that are equivalent to those obtained using previously developed mesh refinement methods for optimal control problems whose solutions are smooth.

Optimal control of renewable biohydrogen production: A switched system approach*

Biohydrogen produced from microorganisms such as cyanobacteria is a promising low cost, sustainable and environmentally friendly energy source. Recent studies have shown that high biohydrogen yield can be obtained from Cyanothece sp. ATCC 51142 in a fed-batch reactor. This system has been accurately described with a modified Droop model that can be used for optimization studies. Searching for the optimal operating conditions and the switching time from batch to fed-batch operation, such that the biohydrogen production is maximized, leads to a challenging singular optimal control problem. In this study, a novel reformulation based on the theory of switched systems and time-scaling transformation is proposed to address the switching of the operating modes and the optimal control structure. Solutions are found by solving an embedded optimal control problem that can be solved efficiently as a nonlinear programming problem. No mesh refinement is required to capture the switching times. Smooth optimal control profiles and clear switching structures that maximize the biohydrogen yield were found for two types of control parametrization.

Optimal installation of renewable electricity sources: the case of Italy

Starting from the model in Koch-Vargiolu (2019), we test the real impact of current renewable installed power in the electricity price in Italy, and assess how much the renewable installation strategy which was put in place in Italy deviated from the optimal one obtained from the model in the period 2012--2018. To do so, we consider the Ornstein-Uhlenbeck (O-U) process, including an exogenous increasing process influencing the mean reverting term, which is interpreted as the current renewable installed power. Using real data of electricity price, photovoltaic and wind energy production from the six main Italian price zones, we estimate the parameters of the model and obtain quantitative results, such as the production of photovoltaic energy impacts the North zone, while wind is significant for Sardinia and the Central North zone does not present electricity price impact. Then we implement the solution of the singular optimal control problem of installing renewable power production devices, in order to maximize the profit of selling the produced energy in the market net of installation costs. We extend the results of \cite{KV} to the case when no impact on power price is presented, and to the case when $N$ players can produce electricity by installing renewable power plants. We are thus able to describe the optimal strategy and compare it with the real installation strategy that was put in place in Italy.

A computational approach to optimal control problems subject to mixed control-state constraints

In this paper, we study a computational approach to Hamilton–Jacobi–Bellman (HJB) equation for singular optimal control subject to mixed control-state constraints. A nonlinear diffusion equation is presented for the viscosity approximation to a partial differential equation by rewriting HJB equation. We construct a so called extremal flow for approximating the optimal objective value of the singular optimal control problem by a differential-algebraic equation. Some examples are given to illustrate this computational approach.

Optimization of polyhydroxyalkanoates bioproduction, based on a cybernetic mathematical model

This work discusses the optimization of the biopolymer PHAs production by Ralstonia eutropha, in a bioreactor carried out under fed-batch mode. Although the optimization of fed-batch fermentations involves the manipulation of the substrate feed rate, which generates a singular optimal control problem, the optimal trajectory can be also set by adjusting small segments by non-linear programming. The cybernetic structured mathematical model used here in is described by a system of 12 differential equations; the strategy involves the maximization/minimization of an Objective Function considering the model as a set of implicit constraints and the discretization of the manipulated variables (substrate feed rates). The sequential quadratic program method is used to solve the optimization problem. PHAs productivity is taken as the objective function and its results are compared to those documented in the literature.

Optimal Impulsive Control Problems Motivated by Mechanical Systems With Vibrations and Blockable DOFs

We consider singular optimal control problems stated for differential-algebraic measure-driven equations. The latter take the form of complementarity systems, which admit jumps of state trajectories produced by open-loop and state-conditioned impulses, acting independently. As an inspiration for the addressed class of models, we turn to mechanical systems driven by time-dependent holonomic (shock impacts, control vibrations) and/or nonholonomic (impactive blocking/releasing system's degrees of freedom) constraints. For our problems, we adapt an approach based on singular space-time transformation. This approach involves approximation of discontinuous solutions of the impulsive complementarity system by ordinary, physically meaningful control processes. Since the complexity of the addressed problems is too high for the analytical investigation, we focus on their numerical investigation, and propose a general computational method.

Robust Multi-Objective Singular Optimal Control Ofpenicillin Fermentation Process

The determination of optimal feeding profile of fed-batch fermentation requires the solution of a singular optimal control problem. The complexity in obtaining the solution to this singular problem is due to the nonlinear dynamics of the system model, the presence of control variables in linear form and the existence of constraints in both the state and control variables. Traditionally, during the optimization process, uncertainties associated with design variables, control parameters and mathematical model are not considered. In this contribution, a systematic methodology to evaluate uncertainties during the resolution of a singular optimal control problem is proposed. This approach consists of the Multi-objective Optimization Differential Evolution algorithm associated with Effective Mean Concept. The proposed methodology is applied to determine the feed substrate concentration in fed-batch penicillin fermentation process. The robust multi- objective singular optimal control problem consists of maximizing the productivity and minimizing the operation total time.

An indirect approach for singular optimal control problems

In singular optimal control problems, the control is not explicitly defined by the optimality conditions. As a consequence, both direct and indirect methods may fail to find a proper solution. This situation is common when optimizing the operation of some process systems, such as batch and continuous reactors. Some direct strategies that rely on regularization and rigorous or heuristic mesh refinement have been recently proposed. In this work, a simple but flexible nonlinear programing formulation is presented, with moving finite elements based on an indirect approach. It is shown that, despite previously reported drawbacks, indirect methods, when properly formulated, are efficient to solve challenging problems. The proposed formulation strictly satisfies the optimality and Weierstrass-Erdmann corner conditions at discrete points and provides accurate results with short CPU times. Heuristics or sophisticated regularization schemes are not required. Two sets of problems were solved to assess the performance of this strategy.

Singular optimal control problems with recursive utilities of mean‐field type

AbstractIn this paper a second‐order necessary condition for singular optimal controls with recursive utilities of mean‐field control systems is investigated. For our mean‐field singular control problem, the second‐order adjoint system is not a single mean‐field backward stochastic differential equation (BSDE), but a matrix‐valued system, which consists of three classical BSDEs. Based on this observation, a new second‐order expansion of cost functional is proved and also the stochastic maximum principle. Our method is appropriate for a large class of mean‐field singular optimal control problems.

Time-minimum control of quantum purity for 2-level Lindblad equations

We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity. The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.

Stochastic maximum principle for mean-field type singular optimal control problem with discounted cost

In this article, mean-field type stochastic singular optimal control problem with discounted cost is studied over an infinite time interval. The discounted cost makes the cost functional is bounded, which guarantees the existence of optimal control. The control variable has two components as classical and singular control. Moreover the singular control satisfies bounded variation, non-decreasing continuous on the left with right limits. The proposed system is investigated in two different cases, such as without and with delay. In addition, infinite horizon version of stochastic maximum principle is established by using the convex control domain in each case. The obtained theoretical results are applied to optimal harvesting problem and optimal consumption problem.

On numerical methods for singular optimal control problems: An application to an AUV problem

We discuss and compare numerical methods to solve singular optimal control problems by the direct method. Our discussion is illustrated by an Autonomous Underwater Vehicle (AUV) problem with state constraints. For this problem, we test four different approaches to solve numerically our problem via the direct method. After discretizing the optimal control problem we solve the resulting optimization problem with (ⅰ) A Mathematical Programming Language ($ \text{AMPL} $), (ⅱ) the Imperial College London Optimal Control Software ($ \text{ICLOCS} $), (ⅲ) the Gauss Pseudospectral Optimization Software ($ \text{GPOPS} $) as well as with (ⅳ) a new algorithm based on mixed-binary non-linear programming reported in [7]. This algorithm consists on converting the optimal control problem to a Mixed Binary Optimal Control (MBOC) problem which is then transcribed to a mixed binary non-linear programming problem ($ \text{MBNLP} $) problem using Legendre-Radau pseudospectral method. Our case study shows that, in contrast with the first three approaches we test (all relying on $ \text{IPOPT} $ or other numerical optimization software packages like $ \text{KNITRO} $), the $ \text{MBOC} $ approach detects the structure of the AUV's problem without a priori information of optimal control and computes the switching times accurately.

Stochastic maximum principle for mean-field type singular optimal control problem with discounted cost

Stochastic maximum principle for mean-field type singular optimal control problem with discounted cost

Singular Control for Two-Dimensional Goddard Problems Under Various Trajectory Bending Laws

Singular optimal control problems are usually characterized by solutions that are difficult to formulate because direct application of Pontryagin’s minimum principle fails to yield the optimal control. When obtained, the singular control becomes a function of the state and the costate variables. Previous research generally showed partial solutions to the problem. In this study, the singular control is found in closed form, as a function of the state variables, via vanishing of a certain determinant. This is an extension of an existing method for finding singular surfaces. The method is applied to several one-dimensional and two-dimensional Goddard problems under various trajectory bending laws. Numerical results, obtained by direct optimization, validate the analytical results. The method can be applied to other problems containing singular arcs.

Singular optimal control by minimizer flows

This paper studies a computational method to deal with a singular optimal control problem by minimizer flows in a viscosity approximation to the Hamilton–Jacobi–Bellman equation. The boundary of the compact constraint set of control variable is intersected with a class of minimizer flows to yield a Hamiltonian extremal function in rewriting the HJB equation. The analysis properties of the flow are revealed in a global optimization framework. An example on computing a minimizer flow and a Hamiltonian extremal function is presented. An application of the minimizer flow in a non-smooth optimization problem is also mentioned.