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When solving analysis, estimation, and control problems for Partial Differential Equations (PDEs) via computational methods, one must resolve three main challenges: (a) the lack of a universal parametric representation of PDEs; (b) handling unbounded differential operators that appear as parameters; and (c), enforcing auxiliary constraints such as Boundary conditions and…
When solving analysis, estimation, and control problems for Partial Differential Equations (PDEs) via computational methods, one must resolve three main challenges: (a) the lack of a universal parametric representation of PDEs; (b) handling unbounded differential operators that appear as parameters; and (c), enforcing auxiliary constraints such as Boundary conditions and continuity conditions. To address these challenges, an alternative representation of PDEs called the `Partial Integral Equation' (PIE) representation is proposed in this work. Primarily, the PIE representation alleviates the problem of the lack of a universal parametrization of PDEs since PIEs have, at most, $12$ Partial Integral (PI) operators as parameters. Naturally, this also resolves the challenges in handling unbounded operators because PI operators are bounded linear operators. Furthermore, for admissible PDEs, the PIE representation is unique and has no auxiliary constraints --- resolving the last of the $3$ main challenges. The PIE representation for a PDE is obtained by finding a unique unitary map from the states of the PIE to the states of the PDE. This map shows a PDE and its associated PIE have equivalent system properties, including well-posedness, internal stability, and I/O behavior. Furthermore, this unique map also allows us to construct a well-defined dual representation that can be used to solve optimal control problems for a PDE. Using the equivalent PIE representation of a PDE, mathematical and computational tools are developed to solve standard problems in Control theory for PDEs. In particular, problems such as a test for internal stability, Input-to-Output (I/O) $L_2$-gain, $\hinf$-optimal state observer design, and $\hinf$-optimal full state-feedback controller design are solved using convex-optimization and Lyapunov methods for linear PDEs in one spatial dimension. Once the PIE associated with a PDE is obtained, Lyapunov functions (or storage functions) are parametrized by positive PI operators to obtain a solvable convex formulation of the above-stated control problems. Lastly, the methods proposed here are applied to various PDE systems to demonstrate the application.
