Abstract: Singular optimal control problems arise frequently on a broad range of chemical engineering applications. Determination of the accurate dynamic structure of the optimal solution profile and the junctions between optimal nonsingular and singular arcs is essential for operating strategies, as well as equipment designs for many processes. In a previous study (Chen, Shao, and Biegler, AIChE J. 60(3), pp. 966–979, 2014) a nested optimization formulation that finds the optimal mesh distribution and determines exact control profiles for nonsingular optimal control problems without state constraints is developed. This study extends this approach to singular optimal control problems. To satisfy the necessary optimality conditions for singular optimal control problems with a well-defined solution strategy, the overall nonlinear programming formulation resulting from direct transcription is decomposed into inner and outer problems. The key feature of this algorithm is that it converges to a solution that satisfies the discretized Euler-Lagrange equations of the original singular optimal control problem; we prove this under suitable assumptions. This is obtained through the introduction of pseudomultipliers that reconstruct the necessary optimality conditions for singular optimal control in the outer problem. We demonstrate this approach on eight classical singular control problems with known solutions, as well as three larger singular control problems derived from chemical engineering applications. © 2016 American Institute of Chemical Engineers AIChE J, 2016

Abstract: This thesis study focuses on the development of model-based control and optimization for optimal grade transitions in polyethylene solution polymerization processes To meet the operational need of this particular process and to reduce transition time and off-grade production for economic benefit, four major topics are taken into account: 1) model development, 2) optimization formulations and solution strategies, 3) handling uncertainties, and 4) online implementation These four parts cover two layers, the real time optimization layer and the advanced control layer, in the decision-making hierarchy of chemical processes Both of them require detailed mathematical models that are representative of the process and efficient dynamic optimization strategies First, a detailed mathematical model is developed to capture the dynamic behavior of the process This includes time delay models for vapor and liquid recycle streams as well as a reduced, yet accurate, vapor-liquid equilibrium (VLE) model derived from rigorous VLE calculations Next, two optimization formulations, single stage and multistage, are developed to deal with single-value target and specification bands of product properties, respectively The results show significant reduction in grade transition time and off-spec production However, the performance can deteriorate in the presence of uncertainties, disturbances and model mismatch, which calls for robust optimization strategies In our work, a flowchart is proposed and back-off constraints calculated from Monte Carlo simulations are incorporated in the original optimization problemto generate optimal control policies that can be applied at different uncertainty levels As an extension of this work, nonlinear model predictive control and state estimation are then considered An online implementation framework is built up for grade transitions in such processes and can be further extended to other similar processes For dynamic optimization, simultaneous collocation method is applied to discretize the differential-algebraic equations, and the resulting nonlinear programming (NLP) problems are solved using NLP solvers Because of the characteristics of the problem, singular control problems are considered and the influence of regularization is discussed for both offline dynamic optimization and optimization under uncertainty

Abstract: In this paper, we study the minimum time planar tilting maneuver of a spacecraft, from the theoretical as well as from the numerical point of view, with a particular focus on the chattering phenomenon. We prove that there exist optimal chattering arcs when a singular junction occurs. Our study is based on the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F. Borisov. We give sufficient conditions on the initial values under which the optimal solutions do not contain any singular arc, and are bang-bang with a finite number of switchings. Moreover, we implement sub-optimal strategies by replacing the chattering control with a fixed number of piecewise constant controls. Numerical simulations illustrate our results.

Abstract: We study PDE of the form $\max\{F(D^2u,x)-f(x), H(Du)\}=0$ where $F$ is uniformly elliptic and convex in its first argument, $H$ is convex, $f$ is a given function and $u$ is the unknown. These equations are derived from dynamic programming in a wide class of stochastic singular control problems. In particular, examples of these equations arise in mathematical finance models involving transaction costs, in queuing theory, and spacecraft control problems. The main aspects of this work are to identify conditions under which solutions are uniquely defined and have Lipschitz continuous gradients. We also generalize previous results known for the case where $M\mapsto F(M,x)$ is the maximum of finitely many linear functions.

Abstract: We prove a verification theorem for a class of singular control problems which model optimal harvesting with density-dependent prices or optimal dividend policy with capital-dependent utilities. The result is applied to solve explicitly some examples of such optimal harvesting/optimal dividend problems. In particular, we show that if the unit price decreases with population density, then the optimal harvesting policy may not exist in the ordinary sense, but can be expressed as a “chattering policy”, i.e. the limit as \(\Delta x\) and \(\Delta t\) go to \(0\) of taking out a sequence of small quantities of size \(\Delta x\) within small time periods of size \(\Delta t\).

Abstract: This paper is concerned with partial-information mixed optimal stochastic continuous–singular control problem for mean-field stochastic differential equation driven by Teugels martingales and an independent Brownian motion, where the Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Levy processes. The control variable has two components; the first being absolutely continuous, and the second singular. Partial-information necessary and sufficient conditions of optimal continuous–singular control for these mean-field models are investigated. As an illustration, this paper studies a partial-information linear quadratic control problem of mean-field type involving continuous–singular control.

Abstract: We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton-Jacobi-Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund's value at a time-dependent optimal boundary. Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at $0$ and created at a rate proportional to its local time.

Abstract: Therapeutic strategies to correct an excessive immune response to pathogenic infection is investigated as an optimal control problem The control problem is formulated around a four dimensional mathematical model describing the inflammatory response to a pathogenic insult with two therapeutic control inputs which have either a direct pro- or anti-inflammatory effect in the given system We use Pontryagin's maximum principle and discuss necessary optimality conditions We consider both an $L^1$ type objective functional as well as an $L^2$ type objective For the former, the presence of singular control will be addressed For each case, numerical simulations using a nonlinear programming optimization solver to acquire different drug treatment strategies are presented and discussed The results provide insight for possible treatment strategies and the methods could be a relevant tool for future practice to assist in better prediction of clinical outcomes and subsequently better treatment for patients

Abstract: We discuss the optimization of chemotherapy treatment for low-grade gliomas using a mathematical model. We analyze the dynamics of the model and study the stability of solutions. The dynamical model is incorporated into an optimal control problem for which different objective functionals are considered. We establish the existence of optimal controls and give a detailed discussion of the necessary optimality conditions. Since the control variable appears linearly in the control problem, optimal controls are concatenations of bang-bang and singular arcs. We derive a formula of the singular control in terms of state and adjoint variables. Using discretization and optimization methods we compute optimal drug protocols in a number of scenarios. For small treatment periods, the optimal control is bang-bang, whereas for larger treatment periods we obtain both bang-bang and singular arcs. In particular, singular controls illustrate the metronomic chemotherapy.

Abstract: In this paper we study a singular control problem for a system of PDEs describing a phase-field model of Penrose−Fife type. The main novelty of this contribution consists in the idea of forcing a sharp interface separation between the states of the system by using heat sources distributed in the domain and at the boundary. We approximate the singular cost functional with a regular one, which is based on the Legendre−Fenchel relations. Then, we obtain a regularized control problem for which we compute the first order optimality conditions using an adapted penalization technique. The proof of some convergence results and the passage to the limit in these optimality conditions lead to the characterization of the desired optimal controller.

Abstract: In this paper, the $$\mathrm{L}^1$$ -minimization for the translational motion of a spacecraft in the circular restricted three-body problem (CRTBP) is considered. Necessary conditions are derived by using the Pontryagin Maximum Principle (PMP), revealing the existence of bang-bang and singular controls. Singular extremals are analyzed, recalling the existence of the Fuller phenomenon according to the theories developed in (Marchal in J Optim Theory Appl 11(5):441–486, 1973; Zelikin and Borisov in Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering. Birkhauser, Basal 1994; in J Math Sci 114(3):1227–1344, 2003). The sufficient optimality conditions for the $$\mathrm{L}^1$$ -minimization problem with fixed endpoints have been developed in (Chen et al. in SIAM J Control Optim 54(3):1245–1265, 2016). In the current paper, we establish second-order conditions for optimal control problems with more general final conditions defined by a smooth submanifold target. In addition, the numerical implementation to check these optimality conditions is given. Finally, approximating the Earth-Moon-Spacecraft system by the CRTBP, an $$\mathrm{L}^1$$ -minimization trajectory for the translational motion of a spacecraft is computed by combining a shooting method with a continuation method in (Caillau et al. in Celest Mech Dyn Astron 114:137–150, 2012; Caillau and Daoud in SIAM J Control Optim 50(6):3178–3202, 2012). The local optimality of the computed trajectory is asserted thanks to the second-order optimality conditions developed.

Abstract: We consider a singular control problem with regime switching that arises in problems of optimal investment decisions of cash-constrained firms. The value function is proved to be the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Moreover, we give regularity properties of the value function as well as a description of the shape of the control regions. Based on these theoretical results, a numerical deterministic approximation of the related HJB variational inequality is provided. We finally show that this numerical approximation converges to the value function. This allows us to describe the investment and dividend optimal policies.

Abstract: We solve explicitly a two-dimensional singular control problem of finite fuel type in infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative price impact with stochastic resilience. The optimal control is obtained as a diffusion process reflected at a non-constant free boundary. To solve the variational inequality and prove optimality, we show new results of independent interest on constructive approximations and Laplace transforms of the inverse local times for diffusions reflected at elastic boundaries.

Abstract: We present an approach for pricing a European call option in presence of proportional transaction costs, when the stock price follows a general exponential L\'evy process. The model is a generalization of the celebrated work of Davis, Panas and Zariphopoulou (1993), where the value of the option is obtained using the concept of utility indifference price. This requires to solve two stochastic singular control problems in finite time, satisfying the same Hamilton-Jacobi-Bellman equation and with different terminal conditions. We solve numerically the continuous time optimization problem using the Markov chain approximation method, and consider the underlying stock following an exponential Merton jump-diffusion process. This model takes into account the possibility of portfolio bankruptcy. We show numerical results for the simpler case of an infinitely rich investor, whose probability of default can be ignored. Option prices are obtained for both the writer and the buyer.

Abstract: An optimal control problem with an infinite horizon quadratic cost functional for a linear system with a known additive disturbance is considered. The feature of this problem is that a weight matrix of the control cost in the cost functional is singular. Due to this singularity, the problem can be solved neither by application of the Pontriagin's Maximum Principle, nor using the Hamilton-Jacobi-Bellman equation approach, i.e. this problem is singular. Since the weight matrix of the control cost, being singular, is not in general zero, only a part of the control coordinates is singular, while the others are regular. This problem is solved by a regularization method. Namely, it is associated with a new optimal control problem for the same equation of dynamics. The cost functional in this new problem is the sum of the original cost functional and an infinite horizon integral of the squares of the singular control coordinates with a small positive weight. Due to a smallness of this coefficient, the new problem is a partial cheap control problem. Using a perturbation technique, an asymptotic analysis of this partial cheap control problem is carried out. Based on this analysis, the infimum of the cost functional in the original problem is derived, and a minimizing sequence of state-feedback controls is designed. An illustrative example of a singular trajectory tracking is presented.

Abstract: In this paper we consider a nonlinear optimal control problem with equality endpoint constraints. We introduce a new natural definition of singular control for which we obtain second-order necessary optimality conditions.

Abstract: In this chapter, we consider an optimal control problem with retarded control and study a larger class of singular (in the classical sense) controls. For the optimality of singular controls, the various necessary conditions in the recurrent forms are obtained. These conditions contain also the analogs of Kelly, Koppa-Mayer, Gabasov, and equality-type conditions. While proving the main results, the Legendre polynomials are used as variations of control.

Abstract: We consider a class of singular control problems driven by a double exponential jump diffusion process, which come from the reversible investment problem. In some interesting cases (e.g., the running cost function is given by the so-called Cobb-Douglas production function), we give the explicit solutions to the singular control problem by using the connection between singular control and optimal switching. We solve a collection of consistent optimal switching problems and yield the explicit solution for the singular control problem. We then give an application to a particular inventory control problem in a single random period.

Abstract: We show an example providing a significance in geometric control theory of the existence of the dependence locus of a system of vector fields in particular, the generic appearance of non-trivial singular trajectories embedded in the dependence locus.

Abstract: In astronautics, an important issue is to control the motion of a satellite subject to the gravitation of celestial bodies in such a way that certain performance indices are minimized (or maximized). In the thesis, we are interested in minimizing the L¹-norm of control for the circular restricted three-body problem. The necessary conditions for optimality are derived by using the Pontryagin maximum principle, revealing the existence of bang-bang and singular controls. Singular extremals are analyzed, and the Fuller phenomenon shows up according to the theories developed by Marchal [1] and Zelikin et al. [2, 3]. The controllability for the controlled two-body problem (a degenerate case of the circular restricted three-body problem) with control taking values in a Euclidean ball is addressed first (cf. Chapter 2). The controllability result is readily extended to the three-body problem since the drift vector field of the three-body problem is recurrent. As a result, if the admissible controlled trajectories remain in a fixed compact set, the existence of the solutions of the L¹-minimizaion problem can be obtained by a combination of Filippov theorem (see [4, Chapter 10], e.g.) and a suitable convexification procedure (see, e.g., [5]). In finite dimensions, the L¹-minimization problem is well-known to generate solutions where the control vanishes on some time intervals. While the Pontryagin maximum principle is a powerful tool to identify candidate solutions for L1-minimization problem, it cannot guarantee that the these candidates are at least locally optimal unless sufficient optimality conditions are satisfied. Indeed, it is a prerequisite to establish (as well as to be able to verify) the necessary and sufficient optimality conditions in order to solve the L¹-minimization problem. In this thesis, the crucial idea for establishing such conditions is to construct a parameterized family of extremals such that the reference extremal can be embedded into a field of extremals. Two no-fold conditions for the canonical projection of the parameterized family of extremals are devised. For the scenario of fixed endpoints, these no-fold conditions are sufficient to guarantee that the reference extremal is locally minimizing provided that each switching point is regular (cf. Chapter 3). If the terminal point is not fixed but varies on a smooth submanifold, an extra sufficient condition involving the geometry of the target manifold is established (cf. Chapter 4). Although various numerical methods, including the ones categorized as direct [6, 7], in- direct [5, 8, 9], and hybrid [10], in the literature are able to compute optimal solutions, one cannot expect a satellite steered by the precomputed optimal control (or nominal control) to move on the precomputed optimal trajectory (or nominal trajectory) due to unavoidable perturbations and errors. In order to avoid recomputing a new optimal trajectory once a deviation from the nominal trajectory occurs, the neighboring optimal feedback control, which is probably the most important practical application of optimal control theory [11, Chapter 5], is derived by parameterizing the neighboring extremals around the nominal one (cf. Chapter 5). Since the optimal control function is bang-bang, the neighboring optimal control consists of not only the feedback on thrust direction but also that on switching times. Moreover, a geometric analysis shows that it is impossible to construct the neighboring optimal control once a conjugate point occurs either between or at switching times.

Abstract: In this paper we establish a new connection between a class of 2-player nonzero-sum games of optimal stopping and certain $2$-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover a differential link between the players' value functions holds across the two games.

Abstract: We consider cancer chemotherapy as an optimal control problem with the aim to minimize a combination of the tumor volume and side effects over an a priori specified therapy horizon when the tumor consists of a heterogeneous agglomeration of many subpopulations. The mathematical model, which accounts for different growth and apoptosis rates in the presence of cell densities, is a finite-dimensional approximation of a model originally formulated by Lorz et al. [18,19] and Greene et al. [10,11] with a continuum of possible traits. In spite of an arbitrarily high dimension, for this problem singular controls (which correspond to time-varying administration schedules at less than maximum doses) can be computed explicitly in feedback form. Interestingly, these controls have the property to keep the entire tumor population constant. Numerical computations and simulations that explore the optimality of bang-bang and singular controls are given. These point to the optimality of protocols that combine a full dose therapy segment with a period of lower dose drug administration.

Abstract: In this paper we study the partial hedging of American options in discrete time and complete markets. This problem leads to a stochastic game maximizing over stopping times and minimizing over stochastic processes satisfying a budget constraint. In the first part of the paper we introduce a dual problem maximizing over two-dimensional vectors of randomized stopping times and parametrized by Lagrange multipliers. We show that the value function of the hedging problem is conjugate to the dual problem. The solution to the dual problem for a fixed Lagrange multiplier is given as the solution of a two-dimensional coupled singular control problem under which the American option is decomposed in terms of risk and cost. In the second part of the paper we show, under a Markov assumption, the existence of an optimal deterministic Markov policy with respect to an enlarged state process.

Abstract: A causal modeling for an impulsive system with a state dependent switching surface is defined and analyzed on a new extended real space, denoted as Krylov hyperreals, which is based on the nonstandard analysis (NSA). The recent work of the authors contains the detailed construction of the extended space, and the generalized function on that space. In the first part of the paper, important concepts of NSA, and the suggested function space are reviewed. Next, a new generalization of a continuous but not differentiable function will be defined on the Krylov hyperreals in order to properly define a composition between singular and non differentiable function. By using an analogy to a spring and damper model, the authors suggest an equivalent causal model of the state dependent impulsive system by introducing the powers of singular control in the original continuous dynamics. A motivational example of a bouncing ball moving on a horizontal surface is analyzed to show the effectiveness.

Abstract: Our main concern in this paper is to study mixed regular-singular control problems, where the control variable has two components, the first being absolutely continuous and the second singular. The coefficients of the state process, as well as the running and final costs are random functions, so as the state process is no longer a Markov process. Our main result is to derive necessary conditions for optimality, also known as the Pontriagin stochastic maximum principle by using Malliavin calculus techniques. The adjoint process, which plays a key role in the stochastic maximum principle, is given by means of the Malliavin derivatives of the optimal state process.