Approximate and Mean Approximate Controllability Properties for Hilfer Time-Fractional Differential Equations
- PDF / 554,838 Bytes
- 27 Pages / 439.642 x 666.49 pts Page_size
- 112 Downloads / 246 Views
Approximate and Mean Approximate Controllability Properties for Hilfer Time-Fractional Differential Equations Ernest Aragones1 · Valentin Keyantuo1 · Mahamadi Warma2 Received: 7 April 2020 / Accepted: 29 July 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator AB with compact resolvent on L2 (), where ⊂ RN (N ≥ 1) is a bounded open set. More precisely, we show that if 0 ≤ ν ≤ 1 and 0 < μ ≤ 1, then the system μ,ν
Dt
u + AB u = f χω
in × (0, T ),
(1−ν)(1−μ)
(It
u)(·, 0) = u0
in ,
is approximately controllable in any time T > 0, u0 ∈ and any nonempty open set ω ⊂ and χω is the characteristic function of ω. In addition, if the operator AB has the unique continuation property, then the system is also mean (memory) approximately controllable. The operator AB can be the realization in L2 () of a symmetric, non-negative uniformly elliptic second order operator with Dirichlet or Robin boundary conditions, or the realization in L2 () of the fractional Laplace operator (−)s (0 < s < 1) with the Dirichlet exterior condition, u = 0 in RN \ , or the nonlocal Robin exterior condition, N s u + βu = 0 in RN \ . L2 ()
Keywords Fractional differential equations · Mittag–Leffler function · Existence and regularity of solutions · Approximately controllable · Mean (memory) approximately controllable Mathematics Subject Classification (2010) 93B05 · 26A33 · 35R11
1 Introduction Mathematical control theory is a topic which has been intensively studied during the past decades. The foundations of modern control theory date back to the early works of Bellman Dedicated to Professor Enrique Zuazua on the occasion of his 60th birthday. Mahamadi Warma
[email protected]
Extended author information available on the last page of the article.
E. Aragones et al.
in the context of dynamic programming [8]; Kalman in filtering techniques and the algebraic approach to linear systems [29, 30]; and Pontryagin with the maximum principle for nonlinear optimal control problems [42]. Within the field of controllability, we can mention the famous paper by Fattorini and Russel [19] that introduced the use of biorthogonal sequences to control one-dimensional heat equations, or the survey paper [44] by Russell that collected a wide spectrum of results in controllability and observability for linear hyperbolic and parabolic partial differential equations (PDEs). We also mention L¨u and Zuazua [34] who very recently proved that for PDEs involving fractional-in-time derivatives, null controllability (hence, exact controllability) cannot be achieved. Thus, for such PDEs it makes sense to study their approximate controllability properties. Enrique Zuazua has played a fundamental role with his outstanding contributions in the development of modern controllability
Data Loading...
