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The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation of Gramians of these systems. In this paper, we develop the method and obtain algorithms for computation of the controllability and observability Gramians of continuous linear and bilinear stationary systems with many inputs and one output, based on the method of spectral expansion of the Gramians and the iterative method for computing the bilinear systems Gramians. An important feature of the concept is the idea of separability of the Gramians expansion: separate computation of the scalar and matrix parts of the spectral Gramian expansion reduces the sub-Gramian matrices computation to calculation of numerical sequences of their elements. For the continuous linear systems with one output the method and the algorithm of the spectral decomposition of the controllability Gramian are developed in the form of Xiao matrices. Analytical expressions for the diagonal elements of the Gramian matrices are obtained, and by making use of which the rest of the elements can be calculated. For continuous linear systems with many outputs the spectral decompositions of the Gramians in the form of generalized Xiao matrices are obtained, which allows us to significantly reduce the number of calculations. The obtained results are generalized for continuous bilinear systems with one output. Iterative spectral algorithms for computation of elements of Gramians of these systems are proposed. Examples are given that illustrate theoretical results.

International audience; A general bilinear optimal control problem subject to an infinitedimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated forma ldifferentiation of the Hamilton–Jacobi-Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-handsides.They form the basis for defining a sub-optimal feedback law.The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker-Planck equation is also provided.

We discuss the numerical solution and perturbation theory for the generalized continuous-time Lyapunov equation E*XA+A*XE=−G with a singular matrix E. If this equation has a solution, it is not unique. We generalize a Bartels–Stewart method and a Hammarling method to compute a partial solution of the generalized Lyapunov equation with a special right-hand side. A spectral condition number is introduced and perturbation bounds for such an equation are presented. Numerical examples are given.