Bang-bang Controls Research Articles

Optimal strategy for stabilizing protein folding intermediates.

To manipulate the protein concentration at a certain functional state through chemical stabilizers is crucial for protein-related studies. It not only plays a key role in protein structure analysis and protein folding kinetics, but also affects protein functionality to a large extent and thus has wide applications in medicine, food industry, etc. However, due to concerns about side effects or financial costs of stabilizers, identifying optimal strategies for enhancing protein stability with a minimal amount of stabilizers is of great importance. Here, we prove that either for the fixed terminal time (including both finite and infinite cases) or for the free one, the optimal control strategy for stabilizing the folding intermediates with a linear strategy for stabilizer addition belongs to the class of bang-bang controls. The corresponding optimal switching time is derived analytically, whose phase diagram with respect to several key parameters is explored in detail. The bang-bang control will be broken when nonlinear strategies for stabilizer addition are adopted. Moreover, the above theory is applied to the stabilization of erythropoietin by ten different kinds of chemicals, providing theoretical guidance for the selection and rational usage of stabilizers. Our current study on optimal strategies for protein stabilizers not only offers deep insights into the general picture of protein folding kinetics but also provides valuable theoretical guidance on treatments for protein-related diseases in medicine.

Stability and Genericity of Bang-Bang Controls in Affine Problems

Stability and Genericity of Bang-Bang Controls in Affine Problems

Stability analysis of the Navier–Stokes velocity tracking problem with bang-bang controls

Stability analysis of the Navier–Stokes velocity tracking problem with bang-bang controls

Strong stabilization of damped nonlinear Schrödinger equation with saturation on unbounded domains

We consider the damped nonlinear Schrödinger equation with saturation: i.e., the complex evolution equation contains in its left hand side, besides the potential term V(x)u, a nonlinear term of the form iμu(t,x)/|u(t,x)| for a given parameter μ>0 (arising in optical applications on non-Kerr-like fibers). In the right hand side we assume a given forcing term f(t,x). The important new difficulty, in contrast to previous results in the literature, comes from the fact that the spatial domain is assumed to be unbounded. We start by proving the existence and uniqueness of weak and strong solutions according the regularity of the data of the problem. The existence of solutions with a lower regularity is also obtained by working with a sequence of spaces verifying the Radon-Nikodým property. Concerning the asymptotic behavior for large times we prove a strong stabilization result. For instance, in the one dimensional case we prove that there is extinction in finite time of the solutions under the mere assumption that the L∞-norm of the forcing term f(t,x) becomes less than μ after a finite time. This presents some analogies with the so called feedback bang-bang controls v (here v=−iμu/|u|+f).

Bang-bang optimal controls for a mathematical model of chemo- and immunotherapy in cancer

Bang-bang optimal controls for a mathematical model of chemo- and immunotherapy in cancer

Risk-neutral valuation of GLWB riders in variable annuities

In this paper we propose a model for pricing GLWB variable annuities under a stochastic mortality framework. Our set-up is very general and only requires the Markovian property for the mortality intensity and the asset price processes. The contract value is defined through an optimization problem which is solved by using dynamic programming. We prove, by backward induction, the validity of the bang-bang condition for the set of discrete withdrawal strategies of the model. This result is particularly remarkable as in the insurance literature either the existence of optimal bang-bang controls is assumed or it requires suitable conditions. We assume constant interest rates, although our results still hold in the case of a Markovian interest rate process. We present extensive numerical examples, modelling the mortality intensity as a non mean reverting square root process and the asset price as an exponential Lévy process, and compare the results obtained for different parameters and policyholder behaviours.

Time-optimal control strategies for tungiasis diseases with limited resources

Tungiasis, as an epidermal parasitic skin disease, leads to long-term injury and life threats to humans in resource-limited countries. In this paper, we aim to minimize the epidemic duration of tungiasis disease with limited resources. A human-jigger parasite system with four control policies for humans and jiggers is established. We propose a time-optimal control problem, in which the eradication time and limited resources are involved. By applying the Pontryagin’s minimum principle, we prove that, only bang-bang controls are admitted for all the control policies. Furthermore, a numerical scheme combining the control parametrization scheme with the interior point algorithm is constructed. Finally, numerical simulations are carried out using actual data of humans and jiggers in Brazil. From all the results, we conclude that, there is a trade-off between the optimal eradication time and the effective funding; once the effective funding is properly used, the disease can be controlled effectively; strengthening personal protections, improving treatments for humans and making adulticiding efforts contribute to keeping humans away from the disease.

A generalization of Müntz-Legendre polynomials and its implementation in optimal control of nonlinear fractional delay systems

A hybrid of Müntz-Legendre polynomials (MLPs) and block-pulse functions (BPFs) is defined and carried out to analyze nonlinear fractional optimal control problems consisting of multiple delays. Instead of using the Caputo fractional derivative, an alternative fractional derivative operator is utilized. The primary optimization problem reduces to an alternative optimization one involving unknown parameters. For this purpose, the fractional Legendre-Gauss quadrature formula is utilized for approximating the associated cost functional and the fractional Legendre-Gaussian nodes are taken as the collocation points. Some new aspects of the proposed basis are demonstrated to verify the effectiveness of the proposed approach based on the Müntz-Legendre basis (MLB) against the classical orthogonal bases. The simulation results certify the feasibility and reliability of the proposed method. Numerous fractional control problems, e.g., bang-bang controls and control systems with any irregularities in the structure of control input can be handled successfully by employing the new fractional basis.

Optimal control model of tumor treatment in the context of cancer stem cell.

We consider cancer cytotoxic drugs as an optimal control problem to stabilize a heterogeneous tumor by attacking not the most abundant cancer cells, but those that are crucial in the tumor ecosystem. We propose a mathematical cancer stem cell model that translates the hierarchy and heterogeneity of cancer cell types by including highly structured tumorigenic cancer stem cells that yield low differentiated cancer cells. With respect to the optimal control problem, under a certain admissibility hypothesis, the optimal controls of our problem are bang-bang controls. These control treatments can retain the entire tumor in the neighborhood of an equilibrium. We simulate the bang-bang control numerically and demonstrate that the optimal drug scheduling should be administered continuously over long periods with short rest periods. Moreover, our simulations indicate that combining multidrug therapies and monotherapies is more efficient for heterogeneous tumors than using each one separately.

Optimal harvesting of a predator-prey system with marine reserve

Fisheries in developing countries are often confronted with the problem of excessive harvesting of the marine resources as a result of lax management practices. This study formulates an optimal control problem involving a predator-prey model with partial harvesting of the prey species and a critical biomass level below which the fishery is not economically viable. The region of boundedness and the equilibrium points of the system are determined and discussed. Stability analysis of the model shows that the interior equilibrium point is both locally and globally asymptotically stable. The Bendixson–Dulac criterion establishes that the system does not possess any limit cycles or periodic orbits. Pontryagin’s maximum principle is employed to determine the necessary conditions for optimality of the model. Singularity analysis of the model, with the aid of the generalized Legendre–Clebsch condition, indicates that the existence of singular arcs cannot be ruled out. Therefore, the characterization of the optimal control gives rise to both bang-bang and singular controls. Furthermore, the relationship between the shadow price of fish stock, the shadow price of fishing effort and the price of landed fish as it relates to the optimal reserve area is explored. Numerical simulations are performed to validate the established theoretical results using empirical data on the Ghana sardinella fishery. The study reveals that the fishery can sustain an effort rate of more than twice the maximum sustainable (MSY) rate, provided that 40% of the current fish stock is in a marine reserve and the operational fish population is at least 30% of the carrying capacity of the ecosystem.

Do generalized draw-down times lead to better dividends? A Pontryagin principle-based answer

Abstract In the context of maximizing cumulative dividends under barrier policies, generalized Azéma–Yor (draw-down) stopping times receive increasing attention during these past years. Based on Pontryagin’s maximality principle, we illustrate the necessity of such generalizations under the framework of spectrally negative Markov processes. Roughly speaking, starting from the explicit expression of the optimal value of discounted dividends in terms of the scale functions, we write down the optimality conditions (via Pontryagin’s principle). The use of generalized draw-downs is then quantified through a structure term (linked to the existence of non bang-bang optimal controls). We thoroughly study several classes of Lévy processes (Bertoin, Lévy Processes, vol. 121. Cambridge University Press, 1998; Kyprianou, Fluctuations of Lévy Processes with Applications: Introductory Lectures. Springer Science & Business Media, 2014) constituting the usual models of insurance claims and a particular piece-wise deterministic Markov model (extending the premium rate to reserve-dependent settings). In all these models, we disprove the consistency of the aforementioned structure equation, thus denying the necessity of such generalizations. We end the paper with some heuristics on possible non-trivial cases for general Markov models.

Designing combination therapies using multiple optimal controls

Strategic management of populations of interacting biological species routinely requires interventions combining multiple treatments or therapies. This is important in key research areas such as ecology, epidemiology, wound healing and oncology. Despite the well developed theory and techniques for determining single optimal controls, there is limited practical guidance supporting implementation of combination therapies. In this work we use optimal control theory to calculate optimal strategies for applying combination therapies to a model of acute myeloid leukaemia. We present a versatile framework to systematically explore the trade-offs that arise in designing combination therapy protocols using optimal control. We consider various combinations of continuous and bang-bang (discrete) controls, and we investigate how the control dynamics interact and respond to changes in the weighting and form of the pay-off characterising optimality. We demonstrate that the optimal controls respond non-linearly to treatment strength and control parameters, due to the interactions between species. We discuss challenges in appropriately characterising optimality in a multiple control setting and provide practical guidance for applying multiple optimal controls. Code used in this work to implement multiple optimal controls is available on GitHub.

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.

Bang-Bang Theorem for a Coupled ODE-PDE Control System

An optimal control problem for a system of one dimensional hyperbolic conservation laws is considered. The system is controlled through the boundary condition, which depends on the state of a nonlinear control system. It is proven that the problem has an optimal solution; moreover, it admits e-optimal solutions generated by bang-bang controls. Bibliography: 8 titles. Illustrations: 1 figure.

Optimal control of acute myeloid leukaemia

Acute myeloid leukaemia (AML) is a blood cancer affecting haematopoietic stem cells. AML is routinely treated with chemotherapy, and so it is of great interest to develop optimal chemotherapy treatment strategies. In this work, we incorporate an immune response into a stem cell model of AML, since we find that previous models lacking an immune response are inappropriate for deriving optimal control strategies. Using optimal control theory, we produce continuous controls and bang-bang controls, corresponding to a range of objectives and parameter choices. Through example calculations, we provide a practical approach to applying optimal control using Pontryagin’s Maximum Principle. In particular, we describe and explore factors that have a profound influence on numerical convergence. We find that the convergence behaviour is sensitive to the method of control updating, the nature of the control, and to the relative weighting of terms in the objective function. All codes we use to implement optimal control are made available.

Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy

In this paper, a non-trivial generalization of a mathematical model put forward in [ 35 ] to account for the development of resistance by tumors to chemotherapy is presented. A study of the existence and local stability of the solutions, as well as the ultimate dynamics of the model, is addressed. An analysis of different chemotherapeutical protocols using discretization and optimization methods is carried out. A number of objective functionals are considered and the necessary optimality conditions are provided. Since the control variable appears linearly in the associated problem, optimal controls are concatenations of bang-bang and singular arcs. A formula of the singular control in terms of state and adjoint variables is derived analytically. Bang-bang and singular controls from the numerical simulations are obtained where, in particular, singular controls illustrate the metronomic chemotherapy.

Error Estimates for Space-Time Discretization of Parabolic Time-Optimal Control Problems with Bang-Bang Controls

In this paper a priori error estimates are derived for full discretization (in space and time) of time-optimal control problems. Various convergence results for the optimal time and the control variable are proved under different assumptions. Especially, the case of bang-bang controls is investigated. Numerical examples are provided to illustrate the results.

Stability for bang-bang control problems of partial differential equations

ABSTRACTIn this paper, we investigate solution stability for control problems of partial differential equations with the cost functional not involving the usual quadratic term for the control. We first establish a sufficient optimality condition for the optimal control problems with bang-bang controls. Then we obtain criteria for solution stability for the optimal control problems of bang-bang controls under linear perturbations. We prove Hölder stability of optimal controls in.

Asymptotic harvesting of populations in random environments.

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. To our knowledge, ergodic optimal control has not been used before to study harvesting strategies. However, it is a natural framework because the optimal harvesting strategy will never be such that the population is harvested to extinction-instead the harvested population converges to a unique invariant probability measure. When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold [Formula: see text] such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate. We also show that, if the yield function is [Formula: see text] and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls.

Optimal control for multi-stage and continuous-time linear singular systems

ABSTRACTIn this paper, optimal control problems for multi-stage and continuous-time linear singular systems are both considered. The singular systems are assumed to be regular and impulse-free. First, a recurrence equation is derived according to Bellman's principle of optimality in dynamic programming. Then, by applying the recurrence equation, bang-bang optimal controls for the control problems with linear objective functions subject to two types of multi-stage singular systems are obtained. Second, employing the principle of optimality, a equation of optimality for settling the optimal control problem subject to a class of continuous-time singular systems is proposed. The optimal control problem may become simpler through solving this equation of optimality. Two numerical examples and a dynamic input–output model are presented to show the effectiveness of the results obtained.