This course develops the theory and design methods for nonlinear and optimal control systems. It moves beyond linear models to study equilibria, stability, feedback design, observability, and performance optimization for systems whose dynamics are genuinely nonlinear. The central goal is to understand how to analyze such systems rigorously and then turn that analysis into control laws, estimators, and algorithms that work in practice.

The chapters build in a natural progression. The first part establishes nonlinear system behavior and Lyapunov stability, then uses those tools for feedback design, feedback linearisation, and observer construction. The second part turns to optimal control, starting from the [calculus of variations](/page/Calculus%20of%20Variations) and the Pontryagin Maximum Principle, then moving to dynamic programming, Hamilton-Jacobi-Bellman equations, and viscosity solutions for nonsmooth value functions. The final chapters address constrained and robust control, including model predictive control, disturbance-aware design, and numerical methods, tying the theory to computational case studies and implementation.

# Introduction

This opening chapter sets the scope of the course and fixes the vocabulary used throughout the notes. The first control course treats linear models, transfer functions, controllability, observability, LQR, and Kalman filtering as the central language. Here the same questions are asked for systems whose dynamics may curve, saturate, switch behaviour across regions of state space, or impose hard constraints on feasible motion.

The course has two intertwined themes. Nonlinear control asks how to design feedback laws when superposition is unavailable and local linear models do not capture global behaviour. Optimal control asks how to choose inputs by minimizing a cost, often under differential constraints, endpoint constraints, uncertainty, and computational limits.

## What Changes Beyond Linear Control?

The first question is why linear theory is not enough. Linearization remains one of the main tools, but a nonlinear system may have several equilibria, finite escape time, invariant sets, state constraints, or behaviour determined by terms that vanish in the first derivative. A controller that stabilizes the tangent model may describe only a local picture, while the applied problem may require a large region of attraction or performance guarantees far from equilibrium.

A useful baseline is the controlled ordinary differential equation. It tells us what the state is, what the input is allowed to do, and how a trajectory is generated.

[definition: Controlled Dynamical System] A controlled dynamical system consists of a state space $X \subset \mathbb R^n$, a control set $U \subset \mathbb R^m$, and a map $f: X \times U \to \mathbb R^n$ such that admissible trajectories satisfy \begin{align*} \dot{x}(t) = f(x(t), u(t)). \end{align*} [/definition]

In most of the course, $X$ is an open subset of $\mathbb R^n$ or a constraint set inside $\mathbb R^n$, and admissible controls are [measurable functions](/page/Measurable%20Functions) $u: [0,T] \to U$. This definition is deliberately broad: it includes mechanical systems, vehicles, population models, chemical reactors, and discretized PDE models.

[example: Pendulum With Torque Input] Let $x_1$ be the angle of a pendulum and let $x_2$ be the angular velocity. With torque input $u \in \mathbb R$, damping coefficient $b \ge 0$, gravitational constant $g>0$, and length $\ell>0$, the state model is \begin{align*} \dot{x}_1=x_2. \end{align*} \begin{align*} \dot{x}_2=-\frac{g}{\ell}\sin x_1-bx_2+u. \end{align*} For a constant input $u=\bar u$, an equilibrium $(\bar x_1,\bar x_2)$ must satisfy \begin{align*} 0=\bar x_2. \end{align*} \begin{align*} 0=-\frac{g}{\ell}\sin \bar x_1-b\bar x_2+\bar u. \end{align*} Substituting $\bar x_2=0$ gives \begin{align*} 0=-\frac{g}{\ell}\sin \bar x_1+\bar u. \end{align*} Equivalently, \begin{align*} \sin \bar x_1=\frac{\ell}{g}\bar u. \end{align*} Thus, when $\bar u=0$, every angle $\bar x_1=k\pi$ with $k\in\mathbb Z$ and velocity $\bar x_2=0$ is an equilibrium. The nonlinear vector field is \begin{align*} f(x_1,x_2,u)=\left(x_2,-\frac{g}{\ell}\sin x_1-bx_2+u\right). \end{align*} Its derivatives with respect to the state variables are \begin{align*} \frac{\partial f_1}{\partial x_1}=0,\quad \frac{\partial f_1}{\partial x_2}=1,\quad \frac{\partial f_2}{\partial x_1}=-\frac{g}{\ell}\cos x_1,\quad \frac{\partial f_2}{\partial x_2}=-b. \end{align*} Therefore the linearization at an equilibrium angle $\bar x_1$ sends a perturbation $(\xi_1,\xi_2)$ to \begin{align*} \left(\xi_2,-\frac{g}{\ell}\cos(\bar x_1)\xi_1-b\xi_2\right). \end{align*} At the downward equilibrium $\bar x_1=0$, $\cos 0=1$, so the linearized perturbation equation is \begin{align*} \dot{\xi}_1=\xi_2,\quad \dot{\xi}_2=-\frac{g}{\ell}\xi_1-b\xi_2. \end{align*} At the upright equilibrium $\bar x_1=\pi$, $\cos \pi=-1$, so the linearized perturbation equation is \begin{align*} \dot{\xi}_1=\xi_2,\quad \dot{\xi}_2=\frac{g}{\ell}\xi_1-b\xi_2. \end{align*} The two linearizations differ by the sign of the stiffness term, even though they come from the same nonlinear equation; this is why a torque law designed near one equilibrium cannot be transferred to the other without checking the nonlinear dynamics and the region where the local model is valid. [/example]

The pendulum also illustrates the first recurring design split. We can either design locally, using a tangent approximation near a chosen equilibrium, or design globally, using energy, invariance, geometry, or optimization.

## Feedback, Stability, and Lyapunov Functions

The next question is what it means for a nonlinear controller to succeed. In linear systems, stability is often read from eigenvalues of a matrix. In nonlinear systems, stability is a property of an equilibrium, an invariant set, or a trajectory, and the answer may depend strongly on the initial condition.

A feedback law closes the loop by choosing the input as a function of the current state. This turns the controlled system into an autonomous dynamical system, so stability theory becomes the core diagnostic.

[definition: State Feedback Law] For a controlled system $\dot{x}=f(x,u)$ with state space $X \subset \mathbb R^n$ and control set $U \subset \mathbb R^m$, a state feedback law is a map $k: X \to U$. The associated closed-loop system is \begin{align*} \dot{x}(t) = f(x(t), k(x(t))). \end{align*} [/definition]

The regularity of $k$ matters. Continuous feedback often suffices for stability statements, while discontinuous feedback and sampled-data feedback require more care about the meaning of a solution. Before measuring stability, we need the point or set around which the closed-loop motion is being compared.

[definition: Equilibrium Of A Closed-Loop System] Let $g: X \to \mathbb R^n$ define an autonomous system $\dot{x}=g(x)$ on $X \subset \mathbb R^n$. A point $x^* \in X$ is an equilibrium if \begin{align*} g(x^*) = 0. \end{align*} [/definition]

An equilibrium identifies the target state, but it does not by itself say whether nearby trajectories stay nearby or approach the target. This motivates the following definition: we need a scalar certificate that measures displacement from the equilibrium and can be differentiated along closed-loop motion.

[definition: Lyapunov Candidate] Let $x^*$ be an equilibrium of $\dot{x}=g(x)$ on $X \subset \mathbb R^n$. A Lyapunov candidate near $x^*$ is a continuously differentiable function $V: D \to \mathbb R$, defined on a neighbourhood $D \subset X$ of $x^*$, such that $V(x^*)=0$ and $V(x)>0$ for all $x \in D \setminus \{x^*\}$. [/definition]