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Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory

Guillaume Carlier, Rabah Tahraoui (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This article is devoted to the optimal control of state equations with memory of the form: x ˙ ( t ) = F ( x ( t ) , u ( t ) , 0 + A ( s ) x ( t - s ) d s ) , t > 0 , with initial conditions x ( 0 ) = x , x ( - s ) = z ( s ) , s > 0 . Denoting by y x , z , u the solution of the previous Cauchy problem and: v ( x , z ) : = inf u V { 0 + e - λ s L ( y x , z , u ( s ) , u ( s ) ) d s } where V is a class of admissible controls, we prove that v is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form: λ v ( x , z ) + H ( x , z , x v ( x , z ) ) + D z v ( x , z ) , z ˙ = 0 in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

Hamilton’s Principle with Variable Order Fractional Derivatives

Atanackovic, Teodor, Pilipovic, Stevan (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 70H25, 46F12, 34K37 Dedicated to 80-th birthday of Prof. Rudolf GorenfloWe propose a generalization of Hamilton’s principle in which the minimization is performed with respect to the admissible functions and the order of the derivation. The Euler–Lagrange equations for such minimization are derived. They generalize the classical Euler-Lagrange equation. Also, a new variational problem is formulated in the case when the order of the derivative is defined through a constitutive equation....

Herbivore harvesting and alternative steady states in coral reefs

Ikbal Hossein Sarkar, Joydeb Bhattacharyya, Samares Pal (2021)

Applications of Mathematics

Coral reefs can undergo relatively rapid changes in the dominant biota, a phenomenon referred to as phase shift. Degradation of coral reefs is often associated with changes in community structure towards a macroalgae-dominated reef ecosystem due to the reduction in herbivory caused by overfishing. We investigate the coral-macroalgal phase shift due to the effects of harvesting of herbivorous reef fish by means of a continuous time model in the food chain. Conditions for local asymptotic stability...

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