A zero-sum electromagnetic evader–interrogator differential game with uncertainty

Banks, Harvey Thomas; Hu, Shuhua

doi:10.1080/00036811.2012.667081

Cited by 3 publications

(4 citation statements)

References 37 publications

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“…As an alternative one could pose our problem as a two-player min-max noncooperative differential game such as used for electromagnetic interrogation/counter-interrogation in [6, 7]. In this case strategies or controls for under/over suppression would represent opposing “player” therapy strategies and one could seek formulations of the game (controls) that provide value (i.e., controls where the min-max=max-min— see [6] and the references therein) for the corresponding differential game. In this case one would find the “best” strategies for balancing virus suppression with organ acceptance.…”

Section: Concluding Remarks and Future Research Effortsmentioning

confidence: 99%

Modelling and optimal control of immune response of renal transplant recipients

Banks

Jang

et al. 2012

Journal of Biological Dynamics

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We consider the increasingly important and highly complex immunological control problem: control of the dynamics of immunosuppression for organ transplant recipients. The goal in this problem is to maintain the delicate balance between over-suppression (where opportunistic latent viruses threaten the patient) and under-suppression (where rejection of the transplanted organ is probable). First, a mathematical model is formulated to describe the immune response to both viral infection and introduction of a donor kidney in a renal transplant recipient. Some numerical results are given to qualitatively validate and demonstrate that this initial model exhibits appropriate characteristics of primary infection and reactivation for immunosuppressed transplant recipients. In addition, we develop a computational framework for designing adaptive optimal treatment regimes with partial observations and low frequency sampling, where the state estimates are obtained by solving a second deterministic optimal tracking problem. Numerical results are given to illustrate the feasibility of this method in obtaining optimal treatment regimes with a balance between under-suppression and over-suppression of the immune system.

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Section: Concluding Remarks and Future Research Effortsmentioning

confidence: 99%

Modelling and optimal control of immune response of renal transplant recipients

Banks

Jang

et al. 2012

Journal of Biological Dynamics

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“…Differential games in infinite dimensions have been investigated by Kocan, Soravia, and Świe ֒ ch [55], by Kocan and Soravia [54], by Shaiju [90], by Ghosh and Shaiju [41], by Ramaswamy and Shaiju [87], by Nowakowska and Nowakowski [81], by Świe ֒ ch [98], by Banks and Shuhua [3], and by Vlasenko, Rutkas, and Chikriȋ [106]. Stochastic differential games in infinite dimensions and the related Isaacs equations have been studied by Fleming and Nisio [37], by Nisio [74][75][76][77][78][79], and by Świe ֒ ch [97].…”

Section: Introductionmentioning

confidence: 99%

Path-dependent Hamilton–Jacobi equations in infinite dimensions

Bayraktar

Keller

2018

Journal of Functional Analysis

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We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and stability) for minimax solutions. A particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle. One of the main difficulties, the lack of compactness in infinite-dimensional Hilbert spaces, is circumvented by working with suitable compact subsets of our path space. As an application, our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of (delay) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskiȋ-Subbotin approach similarly as in finite dimensions. 8. Applications to differential games 33 Appendix A. Properties of solution sets of evolution equations 40 Appendix B. Other notions of solutions 45 B.1. Classical solutions 45 B.2. Viscosity solutions 47 References 48 C(S) := {u : S → R continuous with respect to d ∞ }, USC(S) := {u : S → R upper semi-continuous with respect to d ∞ }, LSC(S) := {u : S → R lower semi-continuous with respect to d ∞ }.Clearly, the elements of those function spaces are non-anticipating.2.2. The operator A, related trajectory spaces, and evolution equations. Throughout this work, we fix an operator A : [0, T ] × V → V * and assume that the following hypotheses hold. H(A): (i) The mappings t → A(t, x), v , v ∈ V , are measurable. (ii) Monotonicity: For a.e. t ∈ (0, T ) and every x, y ∈ V , A(t, x) − A(t, y), x − y ≥ 0. (iii) Hemicontinuity: For a.e. t ∈ (0, T ) and every x, y, v ∈ V , the mappings s → A(t, x + sy), v , [0, 1] → R, are continuous.

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“…The resulting problem is formulated as a two-player zero-sum dynamic differential game in which the cost functional is based on the expected value of the intensity of the reflected signal. In [29] we show that there exists a saddle point for the relaxed form of this dynamic differential game in which the relaxed controls appear bilinearly in the dynamics governed by a partial differential equation. We also have developed a computational framework for construction of approximate saddle point strategies in feedback form for a special case of this relaxed differential game with strategies and payoff in the sense of Berkovitz.…”

Section: Dynamic Evasion-interrogation Games With Uncertaintymentioning

confidence: 99%

“…We report new progress [25], [29] on dynamic electromagnetic evasion-interrogation games in which the evader can use ferroelectric material coatings to attempt to avoid detection while the interrogator can manipulate the interrogating frequencies to enhance detection. The resulting problem is formulated as a two-player zero-sum dynamic differential game in which the cost functional is based on the expected value of the intensity of the reflected signal.…”

Section: Dynamic Evasion-interrogation Games With Uncertaintymentioning

confidence: 99%

Fokker-Planck Equations: Uncertainty in Network Security Games and Information

Banks¹,

Ito²

2012

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The public reportmg burd en for th1s collection of mform ation is estimated to average 1 hour per response. includmg the tim e for rev>ewing instruct>ons. searclhing exi sting data so urces. gathering and mmnta1mng the data needed . and co mpleting and rev1ew 1 ng the collection of 1nforma t1 on. Send co mments regarding th1s burden est1m ate or any other aspect o fth1s collection of mformat1on . 1nclud1 ng suggesti ons for reduc1 ng the burden. to the Department of De fense. ExecutiV e Serv1ce D>rectorate (0704 -0 188) Respo ndents should be aware that notwithstanding any other prov1 s1 on of law. no person shall be sub;ect to any penalty for falling to comply w1th a collect> on of 1 nform at1on if 1 t does not display a currently v alid OMB co ntrol number SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S)AF ABSTRACTWe have significant accomplishments on uncertainty quantification in inverse problems for dynamical systems, generalized sensitivity and optimal design of experiments, elasticity and viscoelasticity modeling for buried target detection, and general inverse problem methodology for proliferating populations. Our efforts have continued on development of efficient accurate integration ofFokker Planck equations. Our objective is to analyze and optimize the dynamic behavior of nonlinear stochastic differential equations, especially the stochastic resonance effects based on the probability density function generated by the Fokker Planck equation. Our approach uses the backward characteristic method and the time-splitting integration of the Fokker Planck equation. 1S. SUBJECT TERMS AbstractWe have significant accomplishments on uncertainty quantification in inverse problems for dynamical systems, generalized sensitivity and optimal design of experiments, elasticity and viscoelasticity modeling for buried target detection, and general inverse problem methodology for proliferating populations. Our efforts have continued on development of efficient accurate integration of Fokker Planck equations. Our objective is to analyze and optimize the dynamic behavior of nonlinear stochastic differential equations, especially the stochastic resonance effects based on the probability density function generated by the Fokker Planck equation. Our approach uses the backward characteristic method and the time-splitting integration of the Fokker Planck equation.

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A zero-sum electromagnetic evader–interrogator differential game with uncertainty

Cited by 3 publications

References 37 publications

Modelling and optimal control of immune response of renal transplant recipients

Modelling and optimal control of immune response of renal transplant recipients

Path-dependent Hamilton–Jacobi equations in infinite dimensions

Fokker-Planck Equations: Uncertainty in Network Security Games and Information

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