Approximate controllability 1 Controllability 1 Controllability problems 1 Fractional Laplacian 1 Fractional heat equation 1 Heat equation 1 Heat transfer 1 Laplace transforms 1 Numerical aspects 1 Numerical methods 1 One-dimensional 1 One-dimensional heat 1 Partial differential equations 1 Penalized HUM 1 Theoretical aspects controllability 1 fractional Laplacian 1 fractional heat equation 1 penalized HUM 1