Exact penalty functions for optimal control problems I: Main theorem and free‐endpoint problems

In this two‐part study, we develop a general approach to the design and analysis of exact penalty functions for various optimal control problems, including problems with terminal and state constraints, problems involving differential inclusions, and optimal control problems for linear evolution equations. This approach allows one to simplify an optimal control problem by removing some (or all) constraints of this problem with the use of an exact penalty function, thus allowing one to reduce optimal control problems to equivalent variational problems and apply numerical methods for solving, eg, problems without state constraints, to problems including such constraints, etc. In the first part of our study, we strengthen some existing results on exact penalty functions for optimisation problems in infinite dimensional spaces and utilise them to study exact penalty functions for free‐endpoint optimal control problems, which reduce these problems to equivalent variational ones. We also prove several auxiliary results on integral functionals and Nemytskii operators that are helpful for verifying the assumptions under which the proposed penalty functions are exact.     

An interior penalty method for optimal control problems with state and input constraints of nonlinear systems

This paper exposes a methodology to solve state and input constrained optimal control problems for nonlinear systems. In the presented ‘interior penalty’ approach, constraints are penalized in a way that guarantees the strict interiority of the approaching solutions. This property allows one to invoke simple (without constraints) stationarity conditions to characterize the unknowns. A constructive choice for the penalty functions is exhibited. The property of interiority is established, and practical guidelines for implementation are given. A numerical benchmark example is given for illustration. © 2014 The Authors. Optimal Control Applications and Methods published by John Wiley & Sons, Ltd.     

Perpetual call options with non‐tradability

We explicitly solve an optimal stopping problem related to the exercise of a perpetual American call option when the option holder cannot trade the underlying asset. We prove the verification theorem for the solution proposed. We derive the moment generating function of the optimal exercise time and also the elasticity of the option value with respect to stock price. The class of admissible utility functions that we solve for contains the CRRA family with some parametric restrictions. This theoretical framework provides the exact exercise boundary and the value of perpetual real options for a self interested manager whose incentives are not aligned with those of the shareholders. It can also serve as an approximation to the valuation of executive stock options. Copyright © 2005 John Wiley & Sons, Ltd.     

工业锅炉燃烧过程的稳态优化模型

建立了一个以能源成本最低为目标、采用神经网络模拟约束条件的工业锅炉燃烧过程稳态优化模型.介绍了该模型的罚函数法求解算法,并给出了一个应用实例.理论计算和实际使用表明,该模型有相当的精度,可以显著提高系统的能源利用率.     

Nerve cell function and synaptic mechanisms

Nerve cells (neurones) are ‘excitable’ cells that can transduce a variety of stimuli into electrical signals, continuously sending information about the external and internal environment (in the form of sequences of action potentials) to the central nervous system (CNS). Interneurones in the CNS integrate this information and send signals along output (efferent) neurones to various parts of the body for the appropriate actions to be taken in response to environmental changes. Networks of neurones have been arbitrarily classified into various nervous systems that gather and transmit sensory information and control skeletal muscle function and autonomic function, etc. The junctions between neurones (synapses) are either electrical or chemical. The former permit the direct transfer of electrical current between cells, whereas the latter utilize chemical signalling molecules (neurotransmitters) to transfer information between cells. Neurotransmitters are mainly amino acids, amines or peptides (although other molecules such as purines and nitric oxide are utilized by some cells), and can be excitatory or inhibitory. Individual neurones within the CNS may receive synaptic inputs from thousands of other neurones. Therefore, each neurone ‘integrates’ this vast complexity of inputs and responds accordingly (either by remaining silent or firing action potentials to other neurones). Adaptations in the function and structure of chemical synapses in particular (synaptic plasticity) are thought to underlie the mechanisms mediating cognitive functions (learning and memory).     

Nerve cell function and synaptic mechanisms

Nerve cells (neurones) are ‘excitable’ cells that can transduce a variety of stimuli into electrical signals, continuously sending information about the external and internal environment (in the form of sequences of action potentials) to the central nervous system (CNS). Interneurones in the CNS integrate this information and send signals along output (efferent) neurones to various parts of the body for the appropriate actions to be taken in response to environmental changes. Networks of neurones have been arbitrarily classified into various nervous systems that gather and transmit sensory information and control skeletal muscle function and autonomic function, etc. The junctions between neurones (synapses) are either electrical or chemical. The former permit the direct transfer of electrical current between cells, whereas the latter utilize chemical signalling molecules (neurotransmitters) to transfer information between cells. Neurotransmitters are mainly amino acids, amines or peptides (although other molecules such as purines and nitric oxide are utilized by some cells), and can be excitatory or inhibitory. Individual neurones within the CNS may receive synaptic inputs from thousands of other neurones. Therefore, each neurone ‘integrates’ this vast complexity of inputs and responds accordingly (either by remaining silent or firing action potentials to other neurones). Adaptations in the function and structure of chemical synapses in particular (synaptic plasticity) are thought to underlie the mechanisms mediating cognitive functions (learning and memory).     

Nerve cell function and synaptic mechanisms

Nerve cells (neurones) are ‘excitable’ cells that can transduce a variety of stimuli into electrical signals, continuously sending information about the external and internal environment (in the form of sequences of action potentials) to the central nervous system (CNS). Interneurones in the CNS integrate this information and send signals along output (efferent) neurones to various parts of the body for the appropriate actions to be taken in response to environmental changes. Networks of neurones have been arbitrarily classified into various nervous systems that gather and transmit sensory information and control skeletal muscle function and autonomic function, etc. The junctions between neurones (synapses) are either electrical or chemical. The former permit the direct transfer of electrical current between cells, whereas the latter utilize chemical signalling molecules (neurotransmitters) to transfer information between cells. Neurotransmitters are mainly amino acids, amines or peptides (although other molecules such as purines and nitric oxide are utilized by some cells), and can be excitatory or inhibitory. Individual neurones within the CNS may receive synaptic inputs from thousands of other neurones. Therefore, each neurone ‘integrates’ this vast complexity of inputs and responds accordingly (either by remaining silent or firing action potentials to other neurones). Adaptations in the function and structure of chemical synapses in particular (synaptic plasticity) are thought to underlie the mechanisms mediating cognitive functions (learning and memory).     

Exact penalization of terminal constraints for optimal control problems

We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm‐minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Interior Penalty Discontinuous Galerkin Based Isogeometric Analysis for Allen-Cahn Equations on Surfaces

We propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, including h-refinement and p-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to the $L^2$ norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in the $L^2$ norm. Numerical tests are given to validate the theory and gauge the good performance of our method.     

Composite Coherent States Approximation for One-Dimensional Multi-Phased Wave Functions

The coherent states approximation for one-dimensional multi-phased wave functions is considered in this paper. The wave functions are assumed to oscillate on a characteristic wave length O(ε) with ε≪1. A parameter recovery algorithm is first developed for single-phased wave function based on a moment asymptotic analysis. This algorithm is then extended to multi-phased wave functions. If cross points or caustics exist, the coherent states approximation algorithm based on the parameter recovery will fail in some local regions. In this case, we resort to the windowed Fourier transform technique, and propose a composite coherent states approximation method. Numerical experiments show that the number of coherent states derived by the proposed method is much less than that by the direct windowed Fourier transform technique.     

Approximation of integro-differential equations associated with piecewise deterministic process

Aim of this paper is to present an approximation scheme for optimal control problems of piecewise deterministic processes and corresponding integro-differential Hamilton–Jacobi–Bellman equations. The method is based on a discrete dynamic programming approach. We discretize the continuous process and the cost functional obtaining a discrete time optimal control problem. The corresponding dynamic programming equation gives an approximation of the integro-differential equation. The main feature of the method is the uniform convergence to the value function of the continuous control problem, which can be characterized as the unique weak solution (in viscosity sense) of the dynamic programming equation. Moreover, under appropriate assumptions, an error estimate on the truncation error is derived. It is worth noting that the method provides approximate feedback controls at any point of the grid without extra computations. An application of the approximation scheme to the numerical solution of an optimal control problem for a storage process is also detailed. © 1997 John Wiley & Sons, Ltd.     

Minimal‐power control of hydrogen evolution reactions

An integral approach to solve finite‐horizon optimal control problems posed by set‐point changes in electrochemical hydrogen reactions is developed. The methodology extends to nonlinear problems with regular, convex Hamiltonians that cannot be explicitly minimized, i.e. where the functional dependence of the H‐minimal control on the state and costate variables is not known. The Lagrangian functions determining trajectory costs will not have special restrictions other than positiveness, but for simplicity the final penalty will be assumed quadratic. The answer to the problem is constructed through the solution to a coupled system of three first‐order quasi‐linear partial differential equations (PDEs) for the missing boundary conditions x(T), λ(0) of the Hamiltonian equations, and for the final value of the control variable u(T). The independent variables of these PDEs are the time‐duration T of the process and the characteristic parameter S of the final penalty. The solution provides information on the whole (T, S)‐family of control problems, which can be used not only to construct the individual optimal control strategies online, but also for global design purposes. Copyright © 2009 John Wiley & Sons, Ltd.     

Autopsies in trauma do not add to peer review or quality assurance

OBJECTIVES: The literature supports the concept that autopsies are useful in uncovering missed injuries or undiagnosed conditions that contribute to death after injury, especially late deaths that take place in the intensive care unit. Review of autopsies are also used as part of the trauma quality assurance (QA) process, and autopsy rates are queried by the American College of Surgeons Committee on Trauma in their reviews. Our hypothesis was that autopsies add little useful clinical or diagnostic information compared with QA peer review analysis in a mature trauma program. METHODS: Autopsies for all mortalities at a Level I trauma center between January 1998 and October 1999 were reviewed. The autopsies were reviewed in a "blinded" fashion such that each review occurred before examination of the chart, the trauma registry, and the findings of the trauma QA peer review. Findings from all sources were compared and examined for Goldman type errors (I-IV). RESULTS: Two hundred sixty-three mortalities were identified, with 216 autopsies reviewed. One hundred two (39%) mortalities were considered dead on arrival to the trauma center by QA review, with no management errors identified (group 1) (Immediate Death group). Sixty-one patients survived more than 48 hours after injury (group 2). One hundred sixty-one (61%) patients were admitted to the hospital. Ninety-nine patients died within 48 hours (Early Death group) and 62 died between 2 and 143 days (Late Death group). There were no Goldman type I errors (major diagnostic discrepancies that might have influenced mortality) identified in either group. Autopsy data did uncover one potentially technical error in a death that was considered nonpreventable on peer review analysis. CONCLUSION: We conclude that autopsy information for either group appeared to add little useful information to the QA peer review of deaths in a mature trauma program. This was true even in cases where the final QA determination was held pending the results of the autopsy. Autopsy rates may not be a useful parameter in evaluating a trauma QA program.     

An improved smoothing technique‐based control vector parameterization method for optimal control problems with inequality path constraints

An improved control vector parameterization (CVP) method is proposed to solve optimal control problems with inequality path constraints by introducing the l₁ exact penalty function and a novel smoothing technique. Both the state and control variables are allowed to appear explicitly in the inequality path constraints simultaneously. By applying the penalty function and smoothing technique, all the inequality path constraints are firstly reformulated as non‐differentiable penalty terms and incorporated into the objective function. Then, the penalty terms are smoothed by using a novel smooth function, leading to a smooth optimal control problem with no inequality path constraints. With discretizing the control space, a corresponding nonlinear programming (NLP) problem is derived, and error between the NLP problem and the original problem is discussed. Results reveal that if the smoothing parameter is sufficiently small, the solution of the NLP problem is approximately equal to the original problem, which shows the convergence of the proposed method. After clarifying some theories of the proposed approach, a concomitant numerical algorithm is put forward with furnishing the updating rules of both the penalty parameter and smoothing parameter. Simulation examples verify the advantages of the proposed method for tackling nonlinear optimal control problems with different inequality path constraints. Copyright © 2016 John Wiley & Sons, Ltd.     

Nerve cell function and synaptic mechanisms

Nerve cells (neurones) are ‘excitable’ cells which can transduce a variety of stimuli into electrical signals, continuously sending information about the external and internal environment (in the form of sequences of action potentials) to the central nervous system (CNS). Interneurones in the CNS integrate this information and send signals along output (efferent) neurones to various parts of the body for the appropriate actions to be taken in response to environmental changes. Networks of neurones have been arbitrarily classified into various nervous systems which gather and transmit sensory information and control skeletal muscle function and autonomic function, etc. The junctions between neurones (synapses) are either electrical or chemical. The former permit the direct transfer of electrical current between cells, whereas the latter utilize chemical signalling molecules (neurotransmitters) to transfer information between cells. Neurotransmitters are mainly amino acids, amines or peptides (although other molecules such as purines and nitric oxide are utilized by some cells), and can be excitatory or inhibitory. Individual neurones within the CNS may receive synaptic inputs from thousands of other neurones. Therefore, each neurone ‘integrates’ this vast complexity of inputs and responds accordingly (either by remaining silent or firing action potentials to other neurones). Adaptations in the function and structure of chemical synapses in particular (synaptic plasticity) are thought to underlie the mechanisms mediating cognitive functions (learning and memory).     

Exact penalty functions for optimal control problems II: Exact penalization of terminal and pointwise state constraints

The second part of our study is devoted to an analysis of the exactness of penalty functions for optimal control problems with terminal and pointwise state constraints. We demonstrate that with the use of the exact penalty function method one can reduce fixed-endpoint problems for linear time-varying systems and linear evolution equations with convex constraints on the control inputs to completely equivalent free-endpoint optimal control problems, if the terminal state belongs to the relative interior of the reachable set. In the nonlinear case, we prove that a local reduction of fixed-endpoint and variable-endpoint problems to equivalent free-endpoint ones is possible under the assumption that the linearized system is completely controllable, and point out some general properties of nonlinear systems under which a global reduction to equivalent free-endpoint problems can be achieved. In the case of problems with pointwise state inequality constraints, we prove that such problems for linear time-varying systems and linear evolution equations with convex state constraints can be reduced to equivalent problems without state constraints, provided one uses the L^∞ penalty term, and Slater's condition holds true, while for nonlinear systems a local reduction is possible, if a natural constraint qualification is satisfied. Finally, we show that the exact L^p-penalization of state constraints with finite p is possible for convex problems, if Lagrange multipliers corresponding to the state constraints belong to L^p′, where p^′ is the conjugate exponent of p, and for general nonlinear problems, if the cost functional does not depend on the control inputs explicitly.     

In vitro culture duration does not impact the ability of encapsulated choroid plexus transplants to prevent neurological deficits in an excitotoxin-lesioned rat model of Huntington's disease

Delivery of neurotrophic molecules to the CNS is a potential treatment strategy for preventing the neuronal loss accompanying many neurological disorders. Choroid plexus (CP) epithelial cells secrete a cocktail of neurotrophic factors, and encapsulated CP transplants are neuroprotective in animal models of stroke and Huntington's disease (HD). Prior to clinical use, it is essential to identify and optimize parameters such as the length of time that transplant products such as encapsulated CP can be maintained. In the present study, neonatal porcine CP was encapsulated within alginate microcapsules and maintained in vitro for 1, 2, or 7 months. The encapsulated cells remained viable (>80%) at all time points and were transplanted unilaterally into the rat striatum. Seven days later, the same animals received unilateral injections of quinolinic acid (QA; 225 nmol) adjacent to the implant site. Separate groups of animals served as controls and received QA alone. After surgery, animals were periodically evaluated for weight loss and were tested for motor function 14 days post-QA. In controls, QA lesions produced a significant loss of body weight and impaired function of the contralateral forelimb. In contrast, implants of CP were potently neuroprotective as rats receiving CP transplants did not lose body weight and were not significantly impaired when tested for motor function. These benefits were independent of the length of time that the cells were held in vitro and demonstrate that the potential potency of alginate encapsulated CP cells can be retained for extremely long periods of time in vitro.     

Association of quadriceps angle with plantar pressure distribution,navicular height and calcaneo-tibial angle

ObjectiveThe aim of study was to analyze the association between Quadriceps Angle (QA) and plantar pressure, navicular height (NH), and calcaneo-tibial angle (CTA).MethodsA total of 64 volunteers (mean age: 22.25 ± 2.54 (range:19–33)) participated in this cross sectional study. EMED-m (Novel GmbH, Germany) electronic pedobarograph was employed for dynamic plantar pressure measurement using two step protocol. The angle between the vertical axis of calcaneus and the long axis of Achilles tendon for CTA. The height of navicular tubercle from the ground was measured while the subject was standing on both feet for NH. QA was measured while the subject was standing in a relaxed posture where both feet bearing equal weight.ResultsThere were significant negative correlations between QA and maximum force (MxF) under the 4th. metatarsal head (MH4). The QA was also significantly correlated with MxF and force-time integral (FTI) under the bigtoe (BT). FTI under the 3rd. metatarsal head (MH3), MH4 and 5th. metatarsal head (MH5) were significantly negatively correlated with QA. Pressure-time integral (PTI) under the MH4 and MH5 were found to be significantly negatively correlated with QA. A significant correlation was also found between QA and NH (p < 0.0001), whilst there was no correlation between QA and CTA. Regression analysis showed that NH was appeared as the major contributor for the QA (β = −0.49, p < 0.001) in the dynamic condition, followed by BT-FTI (β = 0.37, p < 0.001) and MH5-MxF (β = −0.21, p < 0.037).ConclusionThese findings may imply that the NH which can at least be controlled by appropriate shoe inserts may affect QA. This way, loading pattern of both plantar region and whole lower extremity may be altered.Level of evidenceLevel III, Diagnostic Study.     

Finding initial costates in finite‐horizon nonlinear‐quadratic optimal control problems

A procedure for obtaining the initial value of the costate in a regular, finite‐horizon, nonlinear‐quadratic problem is devised in dimension one. The optimal control can then be constructed from the solution to the Hamiltonian equations, integrated on‐line. The initial costate is found by successively solving two first‐order, quasi‐linear, partial differential equations (PDEs), whose independent variables are the time‐horizon duration T and the final‐penalty coefficient S. These PDEs need to be integrated off‐line, the solution rendering not only the initial condition for the costate sought in the particular (T, S)‐situation but also additional information on the boundary values of the whole two‐parameter family of control problems, that can be used for design purposes. Results are tested against exact solutions of the PDEs for linear systems and also compared with numerical solutions of the bilinear‐quadratic problem obtained through a power‐series' expansion approach. Bilinear systems are specially treated in their character of universal approximations of nonlinear systems with bounded controls during finite time‐periods. Copyright © 2007 John Wiley & Sons, Ltd.