linearquadraticnothreshold

Linearquadratic models, often referred to as linear-quadratic (LQ) models, are a class of mathematical frameworks used in control theory, operations research, and economics to describe systems governed by quadratic cost functions and linear dynamics. These models are particularly useful for analyzing optimal control problems, where the goal is to determine the best possible control inputs to minimize a given performance criterion.

In a linear-quadratic model, the system dynamics are typically represented by a set of linear differential equations, while the cost function is quadratic in both the state variables and control inputs. The general form of a continuous-time linear-quadratic regulator (LQR) problem involves minimizing a cost function of the form:

J = ∫₀^∞ (xᵀQx + uᵀRu + 2xᵀNu) dt

where x represents the state vector, u the control input, and Q, R, and N are symmetric matrices defining the weighting of state, control, and cross terms, respectively. The solution to this problem often involves solving the algebraic Riccati equation, which yields an optimal feedback control law of the form u = -Kx, where K is the feedback gain matrix.

Linear-quadratic models are widely applied in engineering for stabilizing unstable systems, trajectory optimization, and resource allocation. They are also foundational in reinforcement learning and adaptive control, where they provide a structured approach to learning optimal policies. The simplicity and tractability of LQ problems make them a standard benchmark for more complex control strategies, though their applicability is limited to systems that can be adequately approximated by linear dynamics and quadratic costs. Extensions, such as linear-quadratic-Gaussian (LQG) models, incorporate stochastic disturbances and state estimation, broadening their utility in real-world applications.