The fundamental principle of intertemporal optimization: Consumer behavior under recursive preferences

This paper is the second of two in which the system studied has a plant which consists of two integrators with a saturable control input, and the performance criterion which is to be minimized is integral-error-squared. In the first paper the known optimum controller was shown to satisfy Pontryagin's maximum principle. In the present paper further verification of the optimum system is obtained by showing that the best performance function satisfies Bellman's partial differential equation. It is found that Bellman's equation is even satisfied for points on the switching curve, Results are also given for other performance criteria.

Pontryagin's Maximum Principle has been applied for optimization of secreted proteins from Pichia pastoris fed-batch fermentation. The objective of this work is to maximize the total accumulated product per unit operation time under different given conditions and system constraints. To obtain optimal solutions, an automated curve-fitting software, Table Curve 2D, was employed to construct the necessary mathematical models and solve the complicated functions. In the solution processes, the end of the glycerol batch phase was defined as the initial state of the system, the end of the methanol fed-batch phase as the final state, the cell mass produced along with product accumulated as state variables, and the specific growth rate (mu) as the control variable. Initially, a relationship between the specific production rate (rho) and mu was established. Then, according to Pontryagin's Maximum Principle, the admissible range of mu and its trajectories for the optimal operations were determined. Four representative cases with different combinations of the operation time along with the initial and final states were evaluated. A close correlation was obtained between the predicted values of the model equation with the experimental results from the Pichia pastoris fed-batch fermentations producing secreted alpha-galactosidase. The approaches proposed here greatly simplify the computational processes and validate the optimization strategy as a generalized approach to maximize the yield from fed-batch fermentations.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

Pontryagin's maximum principle for optimal control of the stationary Navier–Stokes equations

The work presented in this thesis concerns the optimisation of the hour-to-hour generation schedule (over a scheduling interval of 24 hours) of the generating stations in an all-hydro electric - system. The day-to-day system storage policy is assumed known from the long-term schedule. System load demand is first treated as deterministic. Later, attempts are made to treat it as stochastic. Throughout the text, system transmission losses are represented by means of the B-coefficient loss formula. Chapter 1 presents brief discussions on various topics which may play - a part in the optimization process. In particular, the concept of frequency control and the distinction between long-term and daily scheduling are outlined. The optimization objective is then formulated and some of the known methods of obtaining the solution are briefly discussed. In chapter 2, the optimization problem is first formulated . as that of minimizing the instantaneous total power drawn from the system subject to satisfying a . predetermined time-varying load demand.This is analogous to the minimization of the instantaneous total cost rate in thermal systems. Station characteristic curves are approximated by straight line segments and head variations are assumed negligible. The optimising - conditions are derived using the Lagrange multiplier technique. A small system consisting of 4 stations is studied, where optimisation is achieved by means of an analogue computer. By introducing weighting factors in the objective function, the storage policy can be satisfied. (Analysis carried out in chapter 3 later reveals that this is the correct measure which must be taken to meet the storage policy). Problems do arise, however, when - there are multi-unit stations present in the system. Due to the non-monotonic nature . of the incremental Output curves' of such stations, a. number of solutions would satisfy the optimising conditions. The optimum solution can only be established after testing each possible solution, a prohibitive task for large systems. Approximating the incremental output curves of such stations may create undesirable errors. Another weakness of the method is that it cannot readily take into account the effects of head variations. The application of Pontryagin's Maximum Principle is attempted in chapter 3. It is shown that, for systems with fixed‚ÄövÑvÆ head stations, the optimising conditions are exactly identical to that derived using the Lagrange multiplier technique with weightings being introduced in the objective function. Unfortunately, those problems associated with the presence of multi‚ÄövÑvÆunit stations are still unsolved. Under ideal system conditions, it is shown that optimum generation schedule would result if all stations; except the frequency control station, were step loaded. Step loading mode is defined as an operating mode where the instantaneous station discharge rate is only allowed to take one of two discrete values with stepwise transitions. The analysis indicates that the switching instants do not play any part, except they must be so chosen that the output of the frequency control station is kept within its limits. When step loading technique is applied to non‚ÄövÑvÆideal systems, the optimisation. becomes that of searching for the optimum switching instants of the step loaded stations. This is discussed in chapter 4. For practical reasons, only two step loading modes are assumed admissible. A system consisting of 18 stations is studied. Experimental results indicate that the optimum step loading modes can be established within - a reasonably short computation time. One striking feature of the step loading technique is that it can handle the optimal scheduling problem where the load demand is treated as being stochastic. The effects of head variations can be estimated fairly accurately. The application of the principle of dynamic programming is discussed in chapter 5. Due to the multi-dimensional nature of the problem, this principle must be combined with the relaxation principle to minimise the computation time and computer storage requirements. A good initial guess for the optimum schedule may be obtained using step loading technique. The step loading modes are then modified by applying the combination of dynamic programming and relaxaxtion principles. Finally, chapter 6 outlines the general conclusions drawn from the work presented in the preceeding chapters. The possibility. of applying Pontryagin's Maximum Principle to obtain the long-term storage policy is also studied. Related future research areas are suggested. To the author's best knowledge, the materials presented in chapters 3, 4 and 5 are original. Chapter 2 is based on the work of Chandler Jr. and Gabrielle, where the mathematical error found in their text has been corrected and the weighting procedure, among other materials has been included.

In this article, the optimal boundary control for a generalised Korteweg–de Vries equation is considered. The Dubovitskii and Milyutin functional analytical approach is adopted in the investigation of Pontryagin's maximum principles of the system in both fixed and free final horizon cases. The necessary conditions are, respectively, presented for the optimal boundary control problems in these two cases.

This paper addresses the problem of time-optional motions for a mobile platform in a planar environment. The platform has two non-steerable independently driven wheels. The overall mission of the robot is expressed in terms of a sequence of via points at which the platform must be at rest in a given configuration (position and orientation). The objective is to plan time-optimal trajectories between these configurations assuming an unobstructed environment. Using Pontryagin's maximum principle (PMP), we formally demonstrate that all time optimal motions of the platform for this problem occur for bang-bang controls on the wheels (at each instant, the acceleration on each wheel is either at its upper or lower limit). The PMP, however, only provides necessary conditions for time optimality. To find the time optimal robot trajectories, we first parameterize the bang-bang trajectories using the switch times on the wheels (the times at which the wheel accelerations change sign). With this parameterization, we can fully search the robot trajectory space and find the switch times that will produce particular paths to a desired final configuration of the platform. We show numerically that robot trajectories with three switch times (two on one wheel, one on the other) can reach any position, while trajectories with four switch times can reach any configuration. By numerical comparison with other trajectories involving similar or greater numbers of switch times, we then identify the sets of time-optimal trajectories. These are uniquely defined using ranges of the parameters, and consist of subsets of trajectories with three switch times for the problem when the final orientation of the robot is not specified, and four switch times when a full final configuration is specified. We conclude with a description of the use of the method for trajectory planning for one of our robots.

<p style='text-indent:20px;'>The optimal harvesting of biological resources, which is directly relevant to sustainable development, has attracted more attention. In this paper, we first prove the existence and uniqueness of generalized solution of a size-stage-structured population model while the optimal harvesting effort is discontinuous. Next, we demonstrate the existence of the optimal harvesting policy. Further, based on the idea of the Pontryagin's maximum principle of the optimal control problem in ordinary differential equations, we derive the maximum principle describing the optimal control. Finally, the dynamical behavior of the population is simulated by solving the corresponding optimality system numerically with an algorithm based on the method of backward Euler implicit finite-difference approximation. The numerical simulations indicate harvesting activity will reduce the quantity of the population and that increasing harvesting cost will result in less adult harvested. This provides guideline of implementing harvesting tactic to guarantee the persistence of the population.</p>

In this paper, the Linear Quadratic Regulator Problem with a positivity constraint on the admissible control set is addressed. Necessary and sufficient conditions for optimality are presented in terms of inner products, projections on closed convex sets, Pontryagin's maximum principle and dynamic programming. The main results are concerned with smoothness of the optimal control and the value function. The maximum principle will be extended to the infinite horizon case. Based on these analytical methods, we propose a numerical algorithm for the computation of the optimal controls for the finite and infinite horizon problem. The numerical methods will be justified by convergence properties between the finite and infinite horizon case on one side and discretized optimal controls and the true optimal control on the other.

Pontryagin's maximum principle is only applicable when right hand side functions in a differential equation of a controlled system and an integrand in criterion function as well as their partial derivatives with respect to x are all continuous for both x and u. The optimal control problem in which any above condition is not satisfied is called a problem with discontinuity. In this case the state space is divided by boundary into some regions in which the maximum principle is applicable. In this paper one approach to find out the optimal control law for the problem is discussed. That is, d optimal backward time trajectories are sought in a whole state space by solving one after another the newly established optimal problem with the maximum principle for each region. Usually, costate vector experiences a jump at the discontinuity point but in our method it arises as the result of the solution and any account for jump condition is not required. The mechanism of jump, however, is also examined in this paper.

We present the complete investigation of the degeneracy phenomenon. We obtain the new version of Pontryagin's maximum principle for problems with state constraints. This maximum principle contains some additional information about the behavior of the Hamiltonian (maximum function): the new jump conditions at the endtimes. It turns out that these jump conditions at the endtimes are the crucial ones in the investigation of the degeneracy phenomenon. Using this result under some natural controllability conditions we obtain the necessary and sufficient conditions for the maximum principle to be informative.

We study the interaction between a network designer and an adversary over a dynamical network. The network consists of nodes performing continuous-time distributed averaging. The goal of the network designer is to assist the nodes reach consensus by changing the weights of a limited number of links in the network. Meanwhile, an adversary strategically disconnects a set of links to prevent the nodes from converging. We formulate two problems to describe this competition where the order in which the players act is reversed in the two problems. We utilize Pontryagin's Maximum Principle (MP) to tackle both problems and derive the optimal strategies. Although the canonical equations provided by the MP are intractable, we provide an alternative characterization for the optimal strategies that makes connection to potential theory. Finally, we provide a sufficient condition for the existence of a saddle-point equilibrium for the underlying zero-sum game.

The use of optimization theory to design simple multivariable control systems

In this paper, we develop an economic growth model taking into account two factors of production: fixed capital and labor force, to study the dynamics of GDP growth. The dependence of the output of these factors is described by a production function of the exponential type. Within the framework of the optimal control theory, the optimization problem for investment levels is being solved to maximize the integral index of consumption. We study the qualitative properties of optimal trajectories as solutions of the Hamiltonian systems arising in Pontryagin's maximum principle. The sensitivity analysis of the equilibrium solutions of the economic system is implemented with respect to the elasticity coefficients of the production function, the depreciation rate of the capital, and the discount factor, and growth trends are indicated. The econometric analysis of the model parameters is provided basing on real data for the Russian economy. In accordance with the results of the regression analysis, the projection of economic development is constructed in conditions of the applicability of the economic growth model.

In this paper, we propose a mathematical model for COVID-19-Associated Pulmonary Aspergillosis (CAPA) co-infection, that enables the study of relationship between prevention and treatment. The next generation matrix is employed to find the reproduction number. We enhanced the co-infection model by incorporating time-dependent controls as interventions based on Pontryagin's maximum principle in obtaining the necessary conditions for optimal control. Finally, we perform numerical experiments with different control groups to assess the elimination of infection. In numerical results, transmission prevention control, treatment controls, and environmental disinfection control provide the best chance of preventing the spread of diseases more rapidly than any other combination of controls.