We address the problem of computing a low-rank estimate Y of the solution X of the Lyapunov equation AX + XA′ + Q = 0 without computing the matrix X itself. This problem has applications in both the reduced-order modeling and the control of large dimensional systems as well as in a hybrid algorithm for the rapid numerical solution of the Lyapunov equation via the alternating direction implicit ...

Prediction and ltering of continuous-time stochastic processes require a solver of a continuous-time di erential Lyapunov equation (cdle). Even though this can be recast into an ordinary di erential equation (ode), where standard solvers can be applied, the dominating approach in Kalman lter applications is to discretize the system and then apply the discrete-time di erence Lyapunov equation (d...

Asymptotic stability of finite-dimensional linear continuous-time periodic (FDLCP) systems is studied by the harmonic analysis. It is first shown that stability can be examined with what we call the harmonic Lyapunov equation. Another necessary and sufficient stability criterion is developed via this generalized Lyapunov equation, which reduces the stability test into that of an approximate FDL...

In a series of papers, we characterized the value function in optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The novelty of our results was that they applied to exit time problems with general nonnegative instantaneous costs, including cases where the instantaneous cost is not uniformly bounded below by positive...

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces. Application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lip...

Fluctuations of the mean field of a globally coupled dynamical systems are discussed. The origin of hidden coherence is related with the instability of the fixed point solution of the self-consistent Perron-Frobenius equation. Collective dynamics in globally coupled tent maps are re-examined, both with the help of direct simulation and the Perron-Frobenius equation. Collective chaos in a single...

Asymptotic stability of finite-dimensional linear continuous-time periodic (FDLCP) systems is studied by the harmonic analysis. It is first shown that stability can be examined with what we call the harmonic Lyapunov equation. Another necessary and sufficient stability criterion is developed via this generalized Lyapunov equation, which reduces the stability test into that of an approximate FDL...

Low gain feedback, a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero, has found several applications in constrained control systems, robust control and nonlinear control. In the continuous-time setting, there are currently three ways of constructing low gain feedback laws the eigenstructure assignment approach, the param...

In a series of papers, we characterized the value function in optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The novelty of our results was that they applied to exit time problems with general nonnegative instantaneous costs, including cases where the instantaneous cost is not uniformly bounded below by positive...

Two Lyapunov functionals are presented for the Enskog equation. One is to describe interactions between particles with various velocities and another is to measure the L distance between two classical solutions. The former yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the verification of the L stability of glob...