In this paper, an adaptive neural network command filter controller is investigated for a class of non-strict feedback stochastic nonlinear systems with full-state constraints. By using the command filter approach and error compensation mechanism, the "explosion of complexity" problem caused by the backstepping method and the filtering errors are eliminated. In order to avoid excessive and burdensome computations and to ensure that the backstepping method works normally for non-strict feedback structures, neural networks are employed to approximate the unknown nonlinear functions that contain all the state variables of the system. Meanwhile, the barrier Lyapunov functions are constructed to ensure the constraints are not transgressed. Finally, based on the Lyapunov stability theorem, an adaptive neural tracking controller is presented to guarantee that all the signals of the closed-loop system are semi-global uniformly ultimately bounded (SGUUB) in probability, and the tracking error converges to a small neighborhood around the origin, besides the full-state constraints are not violated. The simulation results are given to confirm the effectiveness of the proposed control method.
{"title":"Barrier Lyapunov functions-based adaptive neural tracking control for non-strict feedback stochastic nonlinear systems with full-state constraints: A command filter approach","authors":"Parisa Seifi, S. K. H. Sani","doi":"10.3934/mcrf.2022024","DOIUrl":"https://doi.org/10.3934/mcrf.2022024","url":null,"abstract":"In this paper, an adaptive neural network command filter controller is investigated for a class of non-strict feedback stochastic nonlinear systems with full-state constraints. By using the command filter approach and error compensation mechanism, the \"explosion of complexity\" problem caused by the backstepping method and the filtering errors are eliminated. In order to avoid excessive and burdensome computations and to ensure that the backstepping method works normally for non-strict feedback structures, neural networks are employed to approximate the unknown nonlinear functions that contain all the state variables of the system. Meanwhile, the barrier Lyapunov functions are constructed to ensure the constraints are not transgressed. Finally, based on the Lyapunov stability theorem, an adaptive neural tracking controller is presented to guarantee that all the signals of the closed-loop system are semi-global uniformly ultimately bounded (SGUUB) in probability, and the tracking error converges to a small neighborhood around the origin, besides the full-state constraints are not violated. The simulation results are given to confirm the effectiveness of the proposed control method.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"7 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74393080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Duality between controllability and observability of stochastic linear evolution equations with general filtration","authors":"Lu Lin","doi":"10.3934/mcrf.2022056","DOIUrl":"https://doi.org/10.3934/mcrf.2022056","url":null,"abstract":"","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"56 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72753280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Studied here is the Kawahara equation, a fifth order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients we prove that solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the Kawahara system is exponentially stable for $T>T_{min}$. After that, introducing a more general delayed system, and by introducing suitable energies, we show using Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.
{"title":"Stabilization results for delayed fifth-order KdV-type equation in a bounded domain","authors":"R. A. Capistrano-Filho, V. H. Martinez","doi":"10.3934/mcrf.2023004","DOIUrl":"https://doi.org/10.3934/mcrf.2023004","url":null,"abstract":"Studied here is the Kawahara equation, a fifth order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients we prove that solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the Kawahara system is exponentially stable for $T>T_{min}$. After that, introducing a more general delayed system, and by introducing suitable energies, we show using Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"14 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73865705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work focuses on the well-posedness of abstract stochastic linear systems with boundary input delay and unbounded observation operators. We use product spaces and a semigroup approach to reformulate such delay systems into free-delay distributed stochastic systems with unbounded control and observation operators. This gives us the opportunity to use the concept of admissible control and observation operators as well as the concept of Yosida extensions to prove the existence and uniqueness of the solution process and provide an estimation of the observation process in relation to initial conditions and control process. As an example, we consider a stochastic Schrödinger system with input delay.
{"title":"A semigroup approach to stochastic systems with input delay at the boundary","authors":"S. Hadd, Fatima-Zahra Lahbiri","doi":"10.3934/mcrf.2022004","DOIUrl":"https://doi.org/10.3934/mcrf.2022004","url":null,"abstract":"This work focuses on the well-posedness of abstract stochastic linear systems with boundary input delay and unbounded observation operators. We use product spaces and a semigroup approach to reformulate such delay systems into free-delay distributed stochastic systems with unbounded control and observation operators. This gives us the opportunity to use the concept of admissible control and observation operators as well as the concept of Yosida extensions to prove the existence and uniqueness of the solution process and provide an estimation of the observation process in relation to initial conditions and control process. As an example, we consider a stochastic Schrödinger system with input delay.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"6 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76718487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p style='text-indent:20px;'>This paper is devoted to analyzing the controllability to rest of the Gurtin-Pipkin model, which is a class of differential equations with memory terms. The goal is not only to derive the state to vanish at some time but also to require the memory term to vanish at the same time, ensuring that the controlled system is controllable to rest. In order to get rid of the influence of memory, the controllability result is obtained by means of the Fourier type approach and the moment theory.</p>
{"title":"Controllability to rest of the Gurtin-Pipkin model","authors":"Xiuxiang Zhou, Shu Luan","doi":"10.3934/mcrf.2021051","DOIUrl":"https://doi.org/10.3934/mcrf.2021051","url":null,"abstract":"<p style='text-indent:20px;'>This paper is devoted to analyzing the controllability to rest of the Gurtin-Pipkin model, which is a class of differential equations with memory terms. The goal is not only to derive the state to vanish at some time but also to require the memory term to vanish at the same time, ensuring that the controlled system is controllable to rest. In order to get rid of the influence of memory, the controllability result is obtained by means of the Fourier type approach and the moment theory.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"7 3","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138518178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Our framework covers the problems for fractional differential equations. Under some necessary convexity conditions, an optimal control exists, and can be characterized via Fréchet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations are not causal, meaning that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. We obtain a causal state feedback representation of the optimal control via a Fredholm integral equation. Finally, a concrete form of our results for fractional differential equations is presented.
{"title":"Causal state feedback representation for linear quadratic optimal control problems of singular Volterra integral equations","authors":"Shuo Han, Ping-Zong Lin, J. Yong","doi":"10.3934/mcrf.2022038","DOIUrl":"https://doi.org/10.3934/mcrf.2022038","url":null,"abstract":"This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Our framework covers the problems for fractional differential equations. Under some necessary convexity conditions, an optimal control exists, and can be characterized via Fréchet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations are not causal, meaning that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. We obtain a causal state feedback representation of the optimal control via a Fredholm integral equation. Finally, a concrete form of our results for fractional differential equations is presented.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"121 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73050466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a control system describing the interaction of water waves with a partially immersed rigid body constraint to move only in the vertical direction. The fluid is modeled by the shallow water equations. The control signal is a vertical force acting on the floating body. We first derive the full governing equations of the fluid-body system in a water tank and reformulate them as an initial boundary value problem of a first-order evolution system. We then linearize the equations around the equilibrium state and we study its well-posedness. Finally we focus on the reachability and stabilizability of the linear system. Our main result asserts that, provided that the floating body is situated in the middle of the tank, any symmetric waves with appropriate regularity can be obtained from the equilibrium state by an appropriate control force. This implies, in particular, that we can project this system on the subspace of states with appropriate symmetry properties to obtain a reduced system which is approximately controllable and strongly stabilizable. Note that, in general, this system is not controllable (even approximately).
{"title":"Shallow water waves generated by a floating object: A control theoretical perspective","authors":"Pei Su, M. Tucsnak","doi":"10.3934/mcrf.2022048","DOIUrl":"https://doi.org/10.3934/mcrf.2022048","url":null,"abstract":"We consider a control system describing the interaction of water waves with a partially immersed rigid body constraint to move only in the vertical direction. The fluid is modeled by the shallow water equations. The control signal is a vertical force acting on the floating body. We first derive the full governing equations of the fluid-body system in a water tank and reformulate them as an initial boundary value problem of a first-order evolution system. We then linearize the equations around the equilibrium state and we study its well-posedness. Finally we focus on the reachability and stabilizability of the linear system. Our main result asserts that, provided that the floating body is situated in the middle of the tank, any symmetric waves with appropriate regularity can be obtained from the equilibrium state by an appropriate control force. This implies, in particular, that we can project this system on the subspace of states with appropriate symmetry properties to obtain a reduced system which is approximately controllable and strongly stabilizable. Note that, in general, this system is not controllable (even approximately).","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"10881 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90242605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the linear Schrödinger equation, in 1D, on a bounded interval, with Dirichlet boundary conditions and bilinear scalar control. The small-time local exact controllability around the ground state was proved in [5], under an appropriate nondegeneracy assumption. Here, we work under a weaker nondegeneracy assumption and we prove the small-time local exact controllability in projection, around the ground state, with estimates on the control (depending linearly on the target) simultaneously in several spaces. These estimates are obtained at the level of the linearized system, thanks to a new result about trigonometric moment problems. Then, they are transported to the nonlinear system by the inverse mapping theorem, thanks to appropriate estimates of the error between the nonlinear and the linearized dynamics.
{"title":"Local controllability of the bilinear 1D Schrödinger equation with simultaneous estimates","authors":"M'egane Bournissou","doi":"10.3934/mcrf.2022027","DOIUrl":"https://doi.org/10.3934/mcrf.2022027","url":null,"abstract":"We consider the linear Schrödinger equation, in 1D, on a bounded interval, with Dirichlet boundary conditions and bilinear scalar control. The small-time local exact controllability around the ground state was proved in [5], under an appropriate nondegeneracy assumption. Here, we work under a weaker nondegeneracy assumption and we prove the small-time local exact controllability in projection, around the ground state, with estimates on the control (depending linearly on the target) simultaneously in several spaces. These estimates are obtained at the level of the linearized system, thanks to a new result about trigonometric moment problems. Then, they are transported to the nonlinear system by the inverse mapping theorem, thanks to appropriate estimates of the error between the nonlinear and the linearized dynamics.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81404251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we study the Schrödinger equation i∂tψ = −∆ψ + η(t) ∑J j=1 δx=aj(t)ψ on L((0, 1),C) where η : [0, T ] −→ R and aj : [0, T ] −→ (0, 1), j = 1, ..., J . We show how to permute the energy associated to different eigenmodes of the Schrödinger equation via suitable choice of the functions η and aj . To the purpose, we mime the control processes introduced in [17] for a very similar equation where the Dirac potential is replaced by a smooth approximation supported in a neighborhood of x = a(t). We also propose a Galerkin approximation that we prove to be convergent and illustrate the control process with some numerical simulations.
{"title":"Achieving energy permutation of modes in the Schrödinger equation with moving Dirac potentials","authors":"Alessandro Duca, C. Castro","doi":"10.3934/mcrf.2022060","DOIUrl":"https://doi.org/10.3934/mcrf.2022060","url":null,"abstract":"In this work, we study the Schrödinger equation i∂tψ = −∆ψ + η(t) ∑J j=1 δx=aj(t)ψ on L((0, 1),C) where η : [0, T ] −→ R and aj : [0, T ] −→ (0, 1), j = 1, ..., J . We show how to permute the energy associated to different eigenmodes of the Schrödinger equation via suitable choice of the functions η and aj . To the purpose, we mime the control processes introduced in [17] for a very similar equation where the Dirac potential is replaced by a smooth approximation supported in a neighborhood of x = a(t). We also propose a Galerkin approximation that we prove to be convergent and illustrate the control process with some numerical simulations.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"31 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81434130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents computational methods for families of linear systems depending on a parameter. Such a family is called ensemble controllable if for any family of parameter-dependent target states and any neighborhood of it there is a parameter-independent input steering the origin into the neighborhood. Assuming that a family of systems is ensemble controllable we present methods to construct suitable open-loop input functions. Our approach to solve this infinite-dimensional task is based on a combination of methods from the theory of linear integral equations and finite-dimensional control theory.
{"title":"Computation of open-loop inputs for uniformly ensemble controllable systems","authors":"Michael Schonlein","doi":"10.3934/mcrf.2021046","DOIUrl":"https://doi.org/10.3934/mcrf.2021046","url":null,"abstract":"This paper presents computational methods for families of linear systems depending on a parameter. Such a family is called ensemble controllable if for any family of parameter-dependent target states and any neighborhood of it there is a parameter-independent input steering the origin into the neighborhood. Assuming that a family of systems is ensemble controllable we present methods to construct suitable open-loop input functions. Our approach to solve this infinite-dimensional task is based on a combination of methods from the theory of linear integral equations and finite-dimensional control theory.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"46 6 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83137370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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