Analysis of the three dimensional heat conduction in nano- or microscale

The Dual-Phase-Lagging (DPL) equation is formulated as an abstract differential equation. In the absence of a heat source term the DPL equation with homogeneous boundary conditions generates a contraction semigroup. The exact expression of the semigroup is achieved. It is proved that the associated eigenfunctions form a Riesz basis. The stability of semigroup is proved. Moreover, it is also shown that the spectrum of DPL equation contains an interval. This implies that the infinitesimal generator associated to the DPL equation is not a Riesz spectral operator. Therefore, the known test for approximate controllability cannot be used. Several controllability properties are investigated

Optimal control solution for Pennes' equation using strongly continuous semigroup

summary:A distributed optimal control problem on and inside a homogeneous skin tissue is solved subject to Pennes' equation with Dirichlet boundary condition at one end and Rubin condition at the other end. The point heating power induced by conducting heating probe inserted at the tumour site as an unknown control function at specific depth inside biological body is preassigned. Corresponding pseudo-port Hamiltonian system is proposed. Moreover, it is proved that bioheat transfer equation forms a contraction and dissipative system. Mild solution for bioheat transfer equation and its adjoint problem are proposed. Controllability and exponentially stability for the related system is proved. The optimal control problem is solved using strongly continuous semigroup solution and time discretization. Mathematical simulations for a thermal therapy in the presence of point heating power are presented to investigate efficiency of the proposed technique

Optimal boundary control for hyperdiffusion equation

summary:In this paper, we consider the solution of optimal control problem for hyperdiffusion equation involving boundary function of continuous time variable in its cost function. A specific direct approach based on infinite series of Fourier expansion in space and temporal integration by parts for analytical solution is proposed to solve optimal boundary control for hyperdiffusion equation. The time domain is divided into number of finite subdomains and optimal function is estimated at each subdomain to obtain desired state with minimum energy. Proposed method has high flexibility so that decision makers are able to trace optimal control in a prescribed subinterval. The implementation of the theory is presented and the effectiveness of the boundary control is investigated by some numerical examples

Controllability and stability of 3D heat conduction equation in a submicroscale thin film

We obtain a closed form analytic solution for the Dual Phase Lagging equation. This equation is a linear, time-independent partial differential equation modeling the heat distribution in a thin film. The spatial domain is of micrometer and nanometer geometries. We show that the solution is described by a semigroup, and obtain a basis of eigenfunctions. The closure of the set of eigenvalues contains an interval, and so the theory on Riesz spectral operator of Curtain and Zwart cannot be applied directly. The exponential stability and the approximate controllability is shown

Novel Interior Point Algorithms for Solving Nonlinear Convex Optimization Problems

This paper proposes three numerical algorithms based on Karmarkar’s interior point technique for solving nonlinear convex programming problems subject to linear constraints. The first algorithm uses the Karmarkar idea and linearization of the objective function. The second and third algorithms are modification of the first algorithm using the Schrijver and Malek-Naseri approaches, respectively. These three novel schemes are tested against the algorithm of Kebiche-Keraghel-Yassine (KKY). It is shown that these three novel algorithms are more efficient and converge to the correct optimal solution, while the KKY algorithm fails in some cases. Numerical results are given to illustrate the performance of the proposed algorithms

Synchronized Control for Five-Story Building under Earthquake Loads

Synchronized control is implemented for a five-story building under earthquake loads and its capabilities are investigated for protection of building under earthquake. In this regard, we applied H2/LQG control algorithm in form of synchronized control for structural vibration reduction. Simulation results of modeling indicated that not only the provided control is able to reduce the responses of vibrations for the structure, but also it is even capable of supplying the objectives of synchronized control at the same time. Numerical results for uncontrolled, traditional H2/LQG control and synchronized control coupled with H2/LQG algorithm are presented. It is shown that for El Centro and Bam earthquakes the synchronized control is more efficient to reduce damage to the given structures

Optimal Control of Anti-Angiogenesis and Radiation Treatments for Cancerous Tumor: Hybrid Indirect Solver

This paper proposes a real-life volume reduction for cancer cells using optimal doses of radiation and an anti-angiogenic drug. A generalized dynamical system based on the diffusion-consumption equation along with stimulation and inhibition factors is proposed. To achieve continuous and low-dose therapy, the related problem is simulated by an optimal regulator problem mathematically. By combining steepest descent, conjugate gradient, and Armijo techniques, a novel hybrid indirect iterative solver is designed. For accuracy and execution speed, the current solver is compared with an interior-point optimizer and sequential quadratic Hamiltonian methods. Cancer therapy under two different treatment strategies and 24 various versions of the general dynamical system is considered numerically. A comprehensive analysis of the corresponding outcomes is presented. Numerical results and related diagrams are provided