arXiv subjects

Sheetal Dharmatti

Publications and source records attributed to Sheetal Dharmatti.

18 recordsLinked to original sources

arXiv2026

Pointwise Tracking Optimal Control Problem for Cahn Hilliard Navier Stokes system

We study a pointwise tracking optimal control problem for the two-dimensional local Cahn Hilliard Navier Stokes system, which models the evolution of two immiscible, incompressible fluids. The source term in the Cahn Hilliard equation acts as a control, and the cost functional measures the deviation of the phase variable from desired values at a finite set of spatial points over time. This setting reflects realistic applications where only a limited number of sensors are available. We also study a variant of the above pointwise tracking control problem where the cost is incorporated with a terminal time pointwise tracking term. The main mathematical difficulty arises from the low regularity of the cost functional due to the pointwise evaluation of the state variables. We prove the existence of strong solutions, establish the existence of an optimal control, and the differentiability of the control to state mapping. We define the adjoint system using a transposition method to characterise optimal control. Moreover, a first-order necessary optimality condition is derived in terms of the adjoint for both problems. Furthermore, we prove that our analysis can be extended to the case of singular potentials.

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arXiv2026

Local controllability of the Cahn-Hilliard-Burgers' equation around certain steady states

In this article we study the local controllability of the one-dimensional Cahn-Hilliard-Navier-Stokes equation, that is Cahn-Hilliard-Burgers' equation, around a certain steady state using a localized interior control acting only in the concentration equation. To do it, we first linearize the nonlinear equation around the steady state. The linearized system turns out to be a system coupled between second order and fourth order parabolic equations and the control acts in the fourth order parabolic equation. The null controllability of the linearized system is obtained by a duality argument proving an observability inequality. To prove the observability inequality, a new Carleman inequality for the coupled system is derived. Next, using the source term method, it is shown that the null controllability of the linearized system with non-homogeneous terms persists provided the non-homogeneous terms satisfy certain estimates in a suitable weighted space. Finally, using a Banach fixed point theorem in a suitable weighted space, the local controllability of the nonlinear system is obtained.

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arXiv2025

Separation Property for the Nonlocal Cahn Hilliard Brinkman System with Singular Potential and Degenerate Mobility

This work studies the nonlocal Cahn Hilliard Brinkman system, which models the phase separation of a binary fluid in a bounded domain and porous media. We focus on a system with a singular potential namely logarithmic form and a degenerate mobility function. The singular potential introduces challenges due to the blow up of its derivatives near pure phases, while the degenerate mobility complicates the analysis. Our main result is the separation property, which ensures that the solution eventually stays away from the pure phases. We adopt a new method, inspired by the De Giorgi iteration, introduced for the two dimensional Cahn Hilliard equation with constant mobility. This work extends previous results and provides a general approach for proving the separation property for similar systems.

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arXiv2024

Well-posedness of three-dimensional Damped Cahn-Hilliard-Navier-Stokes Equations

This paper presents a mathematical analysis of the evolution of a mixture of two incompressible, isothermal fluids flowing through a porous medium in a three dimensional bounded domain. The model is governed by a coupled system of convective Brinkman Forchheimer equations and the Cahn Hilliard equation, considering a regular potential and non degenerate mobility. We first establish the existence of a Leray Hopf weak solution for the coupled system when the absorption exponent r greater than or equal to 1. Additionally, we prove that every weak solution satisfies the energy equality for greater than 3. This further leads to the uniqueness of weak solutions in three-dimensional bounded domains, subject to certain restrictions on the viscosity and the Forchheimer coefficient in the critical case r=3. Moreover, we provide an alternative simplified proof for the uniqueness of weak solutions for r greater than or equal to 3 that holds without imposing any restrictions on viscosity or Forchheimer coefficient. Similar results are also obtained for the case of degenerate mobility and singular potential.

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arXiv2024

Optimal boundary control for the Cahn-Hilliard-Navier-Stokes Equations

In this work, we study an optimal boundary control problem for a Cahn - Hilliard -Navier-Stokes (CHNS) system in a two dimensional bounded domain. The CHNS system consists of a Navier-Stokes equation governing the fluid velocity field coupled with a convective Cahn - Hilliard equation for the relative concentration of the fluids. An optimal control problem is formulated as the minimization of a cost functional subject to the controlled CHNS system where the control acts on the boundary of the Navier-Stokes equations. We first prove that there exists an optimal boundary control. Then we establish that the control-to-state operator is Frechet differentiable and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint system.

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arXiv2024

Existence and Uniqueness of viscosity solutions of Value function of Local Cahn-Hilliard-Navier-Stokes system

In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation with regular potential in two dimensional bounded domain. We formulate distributed optimal control problem as the minimization of a suitable cost functional subject to the controlled local Cahn-Hilliard-Navier- Stokes system and define the associated value function. We prove the Dynamic Programming Principle satisfied by the value function. Due to the lack of smoothness properties for the value function, we use the method of viscosity solutions to obtain the corresponding solution of the infinite dimensional Hamilton-Jacobi-Bellman equation. We show that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solution is established via comparison principle.

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arXiv2023

Regularity and Optimal Control of Non Local Cahn Hilliard Brinkman system with Singular Potential

The evolution of two incompressible, immiscible, isothermal fluids in a bounded domain and a porous media is described by the coupled Cahn-Hilliard-Brinkman (CHB) system. The CHB system consists of the Cahn-Hilliard equation describing the dynamics of the relative concentration of fluids and the Brinkman equation for velocity. This work addresses the optimal control problem for a two-dimensional nonlocal CHB system with a singular-type potential. The existence and regularity results are obtained by approximating the singular potential by a sequence of regular potentials and introducing a sequence of mobility terms to resolve the blow-up due to the singularity of the potential. Further, we prove the existence of a strong solution under higher regularity assumptions on the initial data and the uniqueness of the solution using the weak-strong uniqueness technique. By considering the external forcing term in the velocity equation as a control, we prove the existence of an optimal control for a tracking type cost functional. The differentiability properties of the control-to-state operator are studied to establish the first-order necessary optimality conditions. Moreover, the optimal control is characterised in terms of the adjoint variable.

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arXiv2022

On Cahn-Hilliard-Navier-Stokes equations with Nonhomogeneous Boundary

The evolution of two isothermal, incompressible, immiscible fluids in a bounded domain is governed by Cahn-Hilliard-Navier-Stokes equations (CHNS System). In this work, we study the well-posedness results for the CHNS system with nonhomogeneous boundary condition for the velocity equation. We obtain the existence of global weak solutions in the two-dimensional bounded domain. We further prove the continuous dependence of the solution on initial conditions and boundary data that will provide the uniqueness of the weak solution. The existence of strong solutions is also established in this work. Furthermore, we show that in the two-dimensional case, each global weak solution converges to a stationary solution.

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arXiv2021

Interior and H$^\infty$ feedback stabilization for sabra Shell model of turbulence

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models along with numerical simulations have been studied earlier. One of the most suitable shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in \cite{PBT}. In this paper we have addressed stabilization problems related to sabra shell model of turbulence. We have studied internal stabilization via finite dimensional controller. Moreover we have also studied optimal robust control problem by solving an infinite time horizon max-min control problem. We first prove the $H^ \infty$ stabilization of the linearized system and charatarize it in terms of a feedback operator by solving an algebric ricatti equation. Finally we show that the control will asymptotically stabilize the nonlinear system.

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arXiv2020

On the Stationary Nonlocal Cahn-Hilliard-Navier-Stokes System: Existence, Uniqueness and Exponential Stability

Cahn-Hilliard-Navier-Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn-Hilliard-Navier-Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder's theorem. Further we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn-Hilliard-Navier-Stokes system for constant mobility parameter and viscosity. Finally, in two dimensions, we establish that the stationary solution is exponentially stable under suitable conditions on mobility parameter and viscosity.

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arXiv2019

Nonlocal Cahn-Hilliard-Brinkman System with regular potential: Regularity and optimal control

In this paper we study optimal control problem for non local Cahn-Hilliard-Brinkman system which models phase separation of binary fluids in porous media. We consider the system in two dimensional bounded domain with regular potential. We extend recently proved existence of weak solution results for such a system and prove the existence of strong solution under certain assumptions on the forcing term and initial datum. Further using our regularity results, we study the tracking type optimal control problem. We prove the existence of an optimal control and establish the first order optimality condition. Lastly, we characterize optimal control in terms of the solution of corresponding adjoint system. The existence of solution for the adjoint system is also established.

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arXiv2019

Pontryagin's maximum principle and second order optimality condition for optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions

In this work, we address some optimal control problems related to the evolution of two isothermal, incompressible, immisible fluids in a two dimensional bounded domain. A distributed optimal control problem is formulated as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn-Hilliard-Navier-Stokes equations. We describe the first order necessary conditions of optimality via Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

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arXiv2019

Maximum Principle and Data Assimilation Problem for the Optimal Control Problems Governed by 2D Nonlocal Cahn-Hillard-Navier-Stokes Equations

We study some optimal control problems associated to the evolution of two isothermal, incompressible, immisible fluids in a two-dimensional bounded domain. The Cahn- Hilliard-Navier-Stokes model consists of a Navier-Stokes equation governing the fluid velocity field coupled with a convective Cahn-Hilliard equation for the relative concentration of one of the fluids. A distributed optimal control problem is formulated as the minimization of a cost functional subject to the controlled nonlocal Cahn-Hilliard- Navier-Stokes equations. We establish the first-order necessary conditions of optimality by proving the Pontryagin's maximum principle for optimal control of such system via the seminal Ekeland variational principle. The optimal control is characterized using the adjoint variable. We also study an another control problem which is similar to data assimilation problems in meteorology of obtaining unknown initial data. Considering the same underlying system as above we establish the optimal initial data in terms of the corresponding adjoint variable.

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arXiv2018

Second Order Optimality Conditions for Optimal Control Problems Governed by 2D Nonlocal Cahn Hilliard Navier Stokes Equations

In this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn-Hilliard-Navier-Stokes equations. We describe the first order necessary conditions of optimality via Pontryagin's minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

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arXiv2017

Control Problems and Invariant Subspaces for the Sabra Shell Model of Turbulence

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models and their existence theory along with numerical simulations have been studied earlier. In this work we study control problems related to sabra shell model of turbulence. We associate two cost functionals: one ensures minimizing turbulence in the system and the other addresses the need of taking the flow near a priori known state. We derive optimal controls in terms of the solution of adjoint equations for corresponding linearized problems. In this work, we also establish feedback controllers which would preserve prescribed physical constraints. Since fluid equations have certain fundamental invariants, we would like to preserve these quantities via a control in the feedback form. We utilize the theory of nonlinear semi groups and represent the feedback control as a multi-valued feedback term which lies in the normal cone of the convex constraint space under consideration.

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arXiv2017

Internal Stabilization of a Class of Parabolic Integro-Differential Equations: Application to Viscoelastic Fluids

In this paper, we prove the stabilizability of abstract Parabolic Integro-Differential Equations (PIDE) in a Hilbert space with decay rate $e^{-γt} $ for certain $γ> 0,$ by means of a finite dimensional controller in the feedback form. We determine a linear feedback law which is obtained by solving an algebraic Riccati equation. To prove the existence of the Riccati operator, we consider a linear quadratic optimal control problem with unbounded observation operator. The abstract theory of stabilization developed here is applied to specific problems related to viscoelastic fluids, e.g. Oldroyd B model and Jeffreys model.

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arXiv2008

Unbounded Viscosity Solutions of Hybrid Control Systems

We study a hybrid control system in which both discrete and continuous controls are involved. The discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set $A$ or a controlled jump set $C$ where controller can choose to jump or not. At each jump, trajectory can move to a different Euclidean space. We allow the cost functionals to be unbounded with certain growth and hence the corresponding value function can be unbounded. We characterize the value function as the unique viscosity solution of the associated quasivariational inequality in a suitable function class. We also consider the evolutionary, finite horizon hybrid control problem with similar model and prove that the value function is the unique viscosity solution in the continuous function class while allowing cost functionals as well as the dynamics to be unbounded.

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arXiv2007

Infinite dimensional differential games with hybrid controls

A two-person zero-sum infinite dimensional differential game of infinite duration with discounted payoff involving hybrid controls is studied. The minimizing player is allowed to take continuous, switching and impulse controls whereas the maximizing player is allowed to take continuous and switching controls. By taking strategies in the sense of Elliott--Kalton, we prove the existence of value and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities.

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