<![CDATA[This paper examines the existence and uniqueness of solutions to Linear Quadratic (LQ) optimal control problems with infinite time horizons in time-varying dynamic systems. By extending Sontag's Theorem to semi-infinite intervals, the properties of The Riccati Differential Equation’s solutions are analyzed under assumptions of essential boundedness and boundedness of the system matrix and cost weights. It is proven that the Riccati matrix solution P(t) exists globally, remains positive definite, and converges to the steady-state limit P∞. The uniqueness of the optimal control–state pair (x,u) is obtained through P(t)-based co-state analysis. Simulations on satellite attitude control systems demonstrate convergence and robustness towards periodic disturbances, supporting applications in adaptive control, robust estimation, and time-varying filtering.]]>
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