System gramians computation for the one and two-dimensional heat equations

• Main Author: Perev, Kamen
• Format: Conference Proceeding
• Language: English
• Subjects: Computation; Controllability; Fourier series; Gramians; Heat transfer; Heat transmission; Initial conditions; Observability (systems); Parabolic differential equations; Thermodynamics
• ISSN: 0094-243X; 1551-7616
• DOI: 10.1063/5.0208348

The paper considers the problem of observability and controllability gramians computation for one-dimensional and two-dimensional heat equations. The heat equation is a parabolic differential equation representing the process of heat flow. In the one-dimensional case, the heat equation models the physical process of a heat flow in a rod. In the two-dimensional case, this equation shows how the heat flow changes in a plate. The solutions of the heat equations are derived by applying the time-space separation principle and by using the Fourier series approximation. The zero input solution participates in computing the observability gramian of the distributed parameter system. The solution of the nonhomogeneous differential equation under zero initial conditions is used for computing the controllability gramian. Both solutions are based on the state space formulation of the infinite dimensional systems into an abstract differential form. The gramians computation procedure uses the Riesz-spectral framework of trajectory representation. The obtained system gramians are derived in explicit form, which allows their simple form of computation.