Manoj Atolia, Prakash Loungani, Helmut Maurer, Willi Semmler
The economy-climate interaction and an appropriate mitigation policy for climate protection have been treated in various types of scientific modeling. Here, we specifically focus on the seminal work by Nordhaus [14, 15] on the economy-climate link. We extend the Nordhaus type model to include optimal policies for mitigation, adaptation and infrastructure investment studying the dynamics of the transition to a low fossil-fuel economy. Formally, the model gives rise to an optimal control problem consisting of a dynamic system with five-dimensional state vector representing stocks of private capital, green capital, public capital, stock of brown energy in the ground, and carbon emissions. The objective function captures preferences over consumption but is also impacted by atmospheric $ mathrm{CO}_2 $ and by mitigation and adaptation policies. Given the numerous challenges to climate change policies the control vector is eight-dimensional comprising mitigation, adaptation and infrastructure investment. Our solutions are characterized by turnpike property and the optimal policies that accomplish the objective of keeping the $ mathrm{CO}_2 $ levels within bound are characterized by a significant proportion of investment in public capital going to mitigation in the initial periods. When initial levels of $ mathrm{CO}_{2} $ are high, adaptation efforts also start immediately, but during the initial period, they account for a smaller proportion of government's public investment.
{"title":"Optimal control of a global model of climate change with adaptation and mitigation","authors":"Manoj Atolia, Prakash Loungani, Helmut Maurer, Willi Semmler","doi":"10.3934/mcrf.2022009","DOIUrl":"https://doi.org/10.3934/mcrf.2022009","url":null,"abstract":"The economy-climate interaction and an appropriate mitigation policy for climate protection have been treated in various types of scientific modeling. Here, we specifically focus on the seminal work by Nordhaus [14, 15] on the economy-climate link. We extend the Nordhaus type model to include optimal policies for mitigation, adaptation and infrastructure investment studying the dynamics of the transition to a low fossil-fuel economy. Formally, the model gives rise to an optimal control problem consisting of a dynamic system with five-dimensional state vector representing stocks of private capital, green capital, public capital, stock of brown energy in the ground, and carbon emissions. The objective function captures preferences over consumption but is also impacted by atmospheric $ mathrm{CO}_2 $ and by mitigation and adaptation policies. Given the numerous challenges to climate change policies the control vector is eight-dimensional comprising mitigation, adaptation and infrastructure investment. Our solutions are characterized by turnpike property and the optimal policies that accomplish the objective of keeping the $ mathrm{CO}_2 $ levels within bound are characterized by a significant proportion of investment in public capital going to mitigation in the initial periods. When initial levels of $ mathrm{CO}_{2} $ are high, adaptation efforts also start immediately, but during the initial period, they account for a smaller proportion of government's public investment.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136297316","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 paper, a finite-horizon optimal control problem involving a dynamical system described by a linear Caputo fractional differential equation and a quadratic cost functional is considered. An explicit formula for the value functional is given, which includes a solution of a certain Fredholm integral equation. A step-by-step feedback control procedure for constructing $varepsilon$-optimal controls with any accuracy $varepsilon>0$ is proposed. The basis for obtaining these results is the study of a solution of the associated Hamilton-Jacobi-Bellman equation with so-called fractional coinvariant derivatives.
{"title":"Value functional and optimal feedback control in linear-quadratic optimal control problem for fractional-order system","authors":"M. Gomoyunov","doi":"10.3934/mcrf.2023002","DOIUrl":"https://doi.org/10.3934/mcrf.2023002","url":null,"abstract":"In this paper, a finite-horizon optimal control problem involving a dynamical system described by a linear Caputo fractional differential equation and a quadratic cost functional is considered. An explicit formula for the value functional is given, which includes a solution of a certain Fredholm integral equation. A step-by-step feedback control procedure for constructing $varepsilon$-optimal controls with any accuracy $varepsilon>0$ is proposed. The basis for obtaining these results is the study of a solution of the associated Hamilton-Jacobi-Bellman equation with so-called fractional coinvariant derivatives.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90233923","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 establish new conditions for obtaining uniform bounds on the moments of discrete-time stochastic processes. Our results require a weak negative drift criterion along with a state-dependent restriction on the sizes of the one-step jumps of the processes. The state-dependent feature of the results make them suitable for a large class of multiplicative-noise processes. Under the additional assumption of Markovian property, new result on ergodicity has also been proved. There are several applications to iterative systems, control systems, and other dynamical systems with state-dependent multiplicative noise, and we include illustrative examples to demonstrate applicability of our results.
{"title":"Moment stability of stochastic processes with applications to control systems","authors":"A. Ganguly, D. Chatterjee","doi":"10.3934/mcrf.2023008","DOIUrl":"https://doi.org/10.3934/mcrf.2023008","url":null,"abstract":"We establish new conditions for obtaining uniform bounds on the moments of discrete-time stochastic processes. Our results require a weak negative drift criterion along with a state-dependent restriction on the sizes of the one-step jumps of the processes. The state-dependent feature of the results make them suitable for a large class of multiplicative-noise processes. Under the additional assumption of Markovian property, new result on ergodicity has also been proved. There are several applications to iterative systems, control systems, and other dynamical systems with state-dependent multiplicative noise, and we include illustrative examples to demonstrate applicability of our results.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78995694","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 expose here a novel application of the so-called coupled complex boundary method – first put forward by Cheng et al. (2014) to deal with inverse source problems – in the framework of shape optimization for solving the exterior Bernoulli problem, a prototypical model of stationary free boundary problems. The idea of the method is to transform the overdetermined problem to a complex boundary value problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary conditions on the free boundary. Then, we optimize the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary. We also prove the existence of the shape derivative of the complex state with respect to the domain. Afterwards, we compute the shape gradient of the cost functional, and characterize its shape Hessian at the optimal domain under a strong, and then a mild regularity assumption on the domain. We then prove the ill-posedness of the proposed shape problem by showing that the latter expression is compact. Also, we devise an iterative algorithm based on a Sobolev gradient scheme via finite element method to solve the minimization problem. Finally, we illustrate the applicability of the method through several numerical examples, both in two and three spatial dimensions.
{"title":"On the new coupled complex boundary method in shape optimization framework for solving stationary free boundary problems","authors":"J. F. Rabago","doi":"10.3934/mcrf.2022041","DOIUrl":"https://doi.org/10.3934/mcrf.2022041","url":null,"abstract":"We expose here a novel application of the so-called coupled complex boundary method – first put forward by Cheng et al. (2014) to deal with inverse source problems – in the framework of shape optimization for solving the exterior Bernoulli problem, a prototypical model of stationary free boundary problems. The idea of the method is to transform the overdetermined problem to a complex boundary value problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary conditions on the free boundary. Then, we optimize the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary. We also prove the existence of the shape derivative of the complex state with respect to the domain. Afterwards, we compute the shape gradient of the cost functional, and characterize its shape Hessian at the optimal domain under a strong, and then a mild regularity assumption on the domain. We then prove the ill-posedness of the proposed shape problem by showing that the latter expression is compact. Also, we devise an iterative algorithm based on a Sobolev gradient scheme via finite element method to solve the minimization problem. Finally, we illustrate the applicability of the method through several numerical examples, both in two and three spatial dimensions.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73829923","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 PDE-ODE model of a flexible satellite that is composed of two identical flexible solar panels and a center rigid body. We prove that the satellite model is exponentially stable in the sense that the energy of the solutions decays to zero exponentially. In addition, we construct two internal model based controllers, a passive controller and an observer based controller, such that the linear and angular velocities of the center rigid body converge to the given sinusoidal signals asymptotically. A numerical simulation is presented to compare the performances of the two controllers.
{"title":"Robust controllers for a flexible satellite model","authors":"T. Govindaraj, Jukka-Pekka Humaloja, L. Paunonen","doi":"10.3934/mcrf.2023007","DOIUrl":"https://doi.org/10.3934/mcrf.2023007","url":null,"abstract":"We consider a PDE-ODE model of a flexible satellite that is composed of two identical flexible solar panels and a center rigid body. We prove that the satellite model is exponentially stable in the sense that the energy of the solutions decays to zero exponentially. In addition, we construct two internal model based controllers, a passive controller and an observer based controller, such that the linear and angular velocities of the center rigid body converge to the given sinusoidal signals asymptotically. A numerical simulation is presented to compare the performances of the two controllers.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81906245","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}
Stephan Gerster, F. Nagel, Aleksey Sikstel, G. Visconti
This work is devoted to the design of boundary controls of physical systems that are described by semilinear hyperbolic balance laws. A computational framework is presented that yields sufficient conditions for a boundary control to steer the system towards a desired state. The presented approach is based on a Lyapunov stability analysis and a CWENO-type reconstruction.
{"title":"Numerical boundary control for semilinear hyperbolic systems","authors":"Stephan Gerster, F. Nagel, Aleksey Sikstel, G. Visconti","doi":"10.3934/mcrf.2022040","DOIUrl":"https://doi.org/10.3934/mcrf.2022040","url":null,"abstract":"This work is devoted to the design of boundary controls of physical systems that are described by semilinear hyperbolic balance laws. A computational framework is presented that yields sufficient conditions for a boundary control to steer the system towards a desired state. The presented approach is based on a Lyapunov stability analysis and a CWENO-type reconstruction.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74605341","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 focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models a stationary incompressible shear-thickening fluid. We start by studying the directional differentiability of the non-smooth term within the state equation as a prior step to demonstrate the directional differentiability of the solution operator. Thereafter, we establish a primal first order necessary optimality condition (Bouligand (B) stationarity), which is derived from the directional differentiability of the solution operator. By using a local regularization of the nonsmooth term and carrying out an asymptotic analysis thereafter, we rigourously derive a weak stationarity system for local minima. By combining the B- and weak stationarity conditions, and using the regularity of the Lagrange multiplier, we are able to obtain a strong stationarity system that includes an inequality for the scalar product between the symmetrized gradient of the state and the Lagrange multiplier.
{"title":"Optimal control of a nonsmooth PDE arising in the modeling of shear–thickening fluids","authors":"J. C. Reyes, Paola Quiloango","doi":"10.3934/mcrf.2023009","DOIUrl":"https://doi.org/10.3934/mcrf.2023009","url":null,"abstract":"This paper focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models a stationary incompressible shear-thickening fluid. We start by studying the directional differentiability of the non-smooth term within the state equation as a prior step to demonstrate the directional differentiability of the solution operator. Thereafter, we establish a primal first order necessary optimality condition (Bouligand (B) stationarity), which is derived from the directional differentiability of the solution operator. By using a local regularization of the nonsmooth term and carrying out an asymptotic analysis thereafter, we rigourously derive a weak stationarity system for local minima. By combining the B- and weak stationarity conditions, and using the regularity of the Lagrange multiplier, we are able to obtain a strong stationarity system that includes an inequality for the scalar product between the symmetrized gradient of the state and the Lagrange multiplier.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76259546","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}
Salah-Eddine Chorfi, G. E. Guermai, L. Maniar, W. Zouhair
The main purpose of this article is to prove a logarithmic convexity estimate for the solution of a linear heat equation subject to dynamic boundary conditions in a bounded convex domain. As an application, we prove the impulsive null approximate controllability for an impulsive heat equation with dynamic boundary conditions.
{"title":"Impulse null approximate controllability for heat equation with dynamic boundary conditions","authors":"Salah-Eddine Chorfi, G. E. Guermai, L. Maniar, W. Zouhair","doi":"10.3934/mcrf.2022026","DOIUrl":"https://doi.org/10.3934/mcrf.2022026","url":null,"abstract":"The main purpose of this article is to prove a logarithmic convexity estimate for the solution of a linear heat equation subject to dynamic boundary conditions in a bounded convex domain. As an application, we prove the impulsive null approximate controllability for an impulsive heat equation with dynamic boundary conditions.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88990130","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 article, we use two case studies from geometry and optimal control of chemical network to analyze the relation between abnormal geodesics in time optimal control, accessibility properties and regularity of the time minimal value function. Introduction. In this article, one considers the time minimal control problem for a smooth system of the form dq dt = f(q, u), where q ∈ M is an open subset of R n and the set of admissible control is the set U of bounded measurable mapping u(·) valued in a control domain U , where U is a two-dimensional manifold of R with boundary. According to the Maximum Principle [14], time minimal solutions are extremal curves satisfying the constrained Hamiltonian equation q̇ = ∂H ∂p , ṗ = − ∂q , H(q, p, u) = M(q, p), (1) where H(q, p, u) = p ·F (q, u) is the pseudo (or non maximized) Hamiltonian, while M(q, p) = maxv∈U H(q, p, u) is the true (maximized) Hamiltonian. A projection of an extremal curve z = (q, p) on the q-space is called a geodesic. Moreover since M is constant along an extremal curve and linear with respect to p, the extremal can be either exceptional (abnormal) if M = 0 or non exceptional if M 6= 0. To refine this classification, an extremal subarc can be either regular if the control belongs to the boundary of U or singular if it belongs to the interior and satisfies the condition ∂H ∂u = 0. Taking q(0) = q0 the accessibility set A(q0, tf ) in time tf is the set ∪u(·)∈U q(tf , x0, u), where t 7→ q(·, q0, u) denotes the solution of the system, with q(0) = q0 and clearly since the time minimal trajectories belongs to the boundary of the accessibility set, the Maximum Principle is a parameterization of this boundary. 2020 Mathematics Subject Classification. Primary: 49K15, 49L99, 53C60, 58K50.
{"title":"Accessibility properties of abnormal geodesics in optimal control illustrated by two case studies","authors":"B. Bonnard, J. Rouot, B. Wembe","doi":"10.3934/mcrf.2022052","DOIUrl":"https://doi.org/10.3934/mcrf.2022052","url":null,"abstract":"In this article, we use two case studies from geometry and optimal control of chemical network to analyze the relation between abnormal geodesics in time optimal control, accessibility properties and regularity of the time minimal value function. Introduction. In this article, one considers the time minimal control problem for a smooth system of the form dq dt = f(q, u), where q ∈ M is an open subset of R n and the set of admissible control is the set U of bounded measurable mapping u(·) valued in a control domain U , where U is a two-dimensional manifold of R with boundary. According to the Maximum Principle [14], time minimal solutions are extremal curves satisfying the constrained Hamiltonian equation q̇ = ∂H ∂p , ṗ = − ∂q , H(q, p, u) = M(q, p), (1) where H(q, p, u) = p ·F (q, u) is the pseudo (or non maximized) Hamiltonian, while M(q, p) = maxv∈U H(q, p, u) is the true (maximized) Hamiltonian. A projection of an extremal curve z = (q, p) on the q-space is called a geodesic. Moreover since M is constant along an extremal curve and linear with respect to p, the extremal can be either exceptional (abnormal) if M = 0 or non exceptional if M 6= 0. To refine this classification, an extremal subarc can be either regular if the control belongs to the boundary of U or singular if it belongs to the interior and satisfies the condition ∂H ∂u = 0. Taking q(0) = q0 the accessibility set A(q0, tf ) in time tf is the set ∪u(·)∈U q(tf , x0, u), where t 7→ q(·, q0, u) denotes the solution of the system, with q(0) = q0 and clearly since the time minimal trajectories belongs to the boundary of the accessibility set, the Maximum Principle is a parameterization of this boundary. 2020 Mathematics Subject Classification. Primary: 49K15, 49L99, 53C60, 58K50.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73515743","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 paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator estimates and a global frequency function argument, which is motivated from a recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.
{"title":"Quantitative unique continuation for parabolic equations with Neumann boundary conditions","authors":"Yueliang Duan, Lijuan Wang, Can Zhang","doi":"10.3934/mcrf.2022058","DOIUrl":"https://doi.org/10.3934/mcrf.2022058","url":null,"abstract":"In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator estimates and a global frequency function argument, which is motivated from a recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75571774","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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