Analysis of the controllability of space-time fractional diffusion - PowerPoint PPT Presentation

Objectives of the talk Space-time fractional order operators Controllability results for space-time fractional PDEs A new notion of boundary control Open problems Analysis of the controllability of space-time fractional diffusion and super diffusion equations Mahamadi Warma (UPR-Rio Piedras) The author is partially supported by the AFOSR Fractional PDEs: Theory, Algorithms and Applications ICERM, 2018 Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators Controllability results for space-time fractional PDEs A new notion of boundary control Open problems Objectives of the talk 1 2 Space-time fractional order operators The fractional Laplace operator Some fractional in time derivatives Controllability results for space-time fractional PDEs 3 The case of nonlocal Schr¨ odinger equations The case of nonlocal wave equations A new notion of boundary control 4 Open problems 5 Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators Controllability results for space-time fractional PDEs A new notion of boundary control Open problems Outline 1 Objectives of the talk Space-time fractional order operators 2 The fractional Laplace operator Some fractional in time derivatives Controllability results for space-time fractional PDEs 3 The case of nonlocal Schr¨ odinger equations The case of nonlocal wave equations 4 A new notion of boundary control 5 Open problems Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators Controllability results for space-time fractional PDEs A new notion of boundary control Open problems The considered problem In this talk we consider the following system of evolution  ∂ α t u ( t , x ) + ( − ∆) s u ( t , x ) = f in Ω × (0 , T ) ,   + Intial conditions , (1.1)  + Boundary conditions .  Here α > 0 is a real number, 0 < s ≤ 1, Ω ⊂ R N is a bounded open set with Lipschitz continuous boundary ∂ Ω, ( − ∆) s is the fractional Laplacian and ∂ α t is a fractional time derivative of Caputo type. If α = 1 (resp. α = 2) we have the heat (resp. wave) equation. If 0 < α < 1 such an equation is said to be of slow diffusion. If 1 < α < 2 then it is said to be of super diffusion. Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators Controllability results for space-time fractional PDEs A new notion of boundary control Open problems Questions How to define the fractional Laplace operator ( − ∆) s ? How to define a time fractional derivative ∂ α t ? Which initial and boundary conditions make the system (1.1) well posed as a Cauchy problem? Is there a function f such that solutions of the system can rest at some time T > 0? In other words, is such system null controllable? Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators The fractional Laplace operator Controllability results for space-time fractional PDEs Some fractional in time derivatives A new notion of boundary control Open problems Outline 1 Objectives of the talk Space-time fractional order operators 2 The fractional Laplace operator Some fractional in time derivatives Controllability results for space-time fractional PDEs 3 The case of nonlocal Schr¨ odinger equations The case of nonlocal wave equations 4 A new notion of boundary control 5 Open problems Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators The fractional Laplace operator Controllability results for space-time fractional PDEs Some fractional in time derivatives A new notion of boundary control Open problems The fractional Laplacian: Using Fourier Analysis Using Fourier analysis, we have that the fractional Laplace operator ( − ∆) s can be defined as the pseudo-differential operator with symbol | ξ | 2 s . That is, ( − ∆) s u = C N , s F − 1 � | ξ | 2 s F ( u ) � , where F and F − 1 denote the Fourier, and the inverse Fourier, transform, respectively, and C ( N , s ) is an appropriate normalizing constant. Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators The fractional Laplace operator Controllability results for space-time fractional PDEs Some fractional in time derivatives A new notion of boundary control Open problems The fractional Laplacian: Using Singular Integrals Let 0 < s < 1 and ε > 0 be real numbers. For a measurable function u : R N → R we let � u ( x ) − u ( y ) ( − ∆) s | x − y | N +2 s dy , x ∈ R N . ε u ( x ) = C N , s { y ∈ R N : | x − y | >ε } The fractional Laplacian ( − ∆) s is defined for x ∈ R N by � u ( x ) − u ( y ) ( − ∆) s u ( x ) = C N , s P.V. ε ↓ 0 ( − ∆) s | x − y | N +2 s dy = lim ε u ( x ) , R N � N +2 s provided that the limit exists, where C N , s := s 2 2 s Γ � 2 . Here Γ N 2 Γ(1 − s ) π denotes the usual Euler-Gamma function. Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators The fractional Laplace operator Controllability results for space-time fractional PDEs Some fractional in time derivatives A new notion of boundary control Open problems The fractional Laplacian: Using the Caffarelli-Silvestre extension Let 0 < s < 1. For u : R N → R in an appropriate space, consider the harmonic extension W : [0 , ∞ ) × R N → R . That is the unique weak solution of the Dirichlet problem � W tt + 1 − 2 s in (0 , ∞ ) × R N , W t + ∆ x W = 0 t (2.1) in R N . W (0 , · ) = u Then the fractional Laplace operator can be defined as t → 0 + t 1 − 2 s W t ( t , x ) , x ∈ R N , ( − ∆) s u ( x ) = − d s lim Γ( s ) where the constant d s is given by d s := 2 2 s − 1 Γ(1 − s ). This is called in the literature, the Caffarelli-Silvestre extension. Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators The fractional Laplace operator Controllability results for space-time fractional PDEs Some fractional in time derivatives A new notion of boundary control Open problems All the definitions coincide Let 0 < s < 1. Then ( − ∆) s u ( x ) = C N , s F − 1 � | ξ | 2 s F ( u ) � � u ( x ) − u ( y ) = C N , s P.V. | x − y | N +2 s dy R N t → 0+ t 1 − 2 s W t ( t , x ) , = − d s lim where we recall that W : [0 , ∞ ) × R N → R is the unique weak solution of the Dirichlet problem (2.1). It is clear that ( − ∆) s is a nonlocal operator. That is, supp[( − ∆) s u ] �⊂ supp[ u ] . Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators The fractional Laplace operator Controllability results for space-time fractional PDEs Some fractional in time derivatives A new notion of boundary control Open problems Derivation of singular integrals: Long jump random walks Let K : R N → [0 , ∞ ) be an even function such that � K ( k ) = 1 . (2.2) k ∈ Z N Given a small h > 0, we consider a random walk on the lattice h Z N . We suppose that at any unit time τ (which may depend on h ) a particle jumps from any point of h Z N to any other point. The probability for which a particle jumps from a point hk ∈ h Z N to the point h ˜ k is taken to be K ( k − ˜ k ) = K (˜ k − k ). Note that, differently from the standard random walk, in this process the particle may experience arbitrarily long jumps, though with small probability. Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs

Objectives of the talk Space-time fractional order operators The fractional Laplace operator Controllability results for space-time fractional PDEs Some fractional in time derivatives A new notion of boundary control Open problems Long jump random walks: Continue Let u ( x , t ) be the probability that our particle lies at x ∈ h Z N at time t ∈ τ Z . Then u ( x , t + τ ) is the sum of all the probabilities of the possible positions x + hk at time t weighted by the probability of jumping from x + hk to x . That is, � K ( k ) u ( x + hk , t ) . u ( x , t + τ ) = k ∈ Z N Using (2.2) we get the following evolution law: � u ( x , t + τ ) − u ( x , t ) = K ( k ) [ u ( x + hk , t ) − u ( x , t )] . (2.3) k ∈ Z N Mahamadi Warma (UPR-Rio Piedras)The author is partially supported by the AFOSR Null Controllability of Fractional PDEs