# What kinds of initial value problems can have superposition of zero state response and zero input response?

I was wondering what kinds of initial value problems can have superposition of zero state response and zero input response?

An example of initial value problem that has the superposition is $$y'(t) = f(t), \quad y(0) = c.$$ But are there other initial value problems that also have such superposition?

Also an initial value problem doesn't necessarily seem to have the superposition in general. An initial value problem is defined as (which is not necessarily first order):

An initial value problem is a differential equation $$y'(t) = f(t, y(t)) \,$$ with $$f: \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$$ where $\Omega \,$ is an open set, together with a point in the domain of $ƒ$ $$(t_0, y_0) \in \Omega,$$ called the initial condition.

A solution to an initial value problem is a function $y$ that is a solution to the differential equation and satisfies $$y(t_0) = y_0. \,$$ This statement subsumes problems of higher order, by interpreting $y$ as a vector. For derivatives of second or higher order, new variables (elements of the vector $y$) are introduced.

I think we also need to specify what an "input" is for an initial value problem.

Thanks and regards!

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## 1 Answer

I find your notation confusing. Therefore, I will denote the state by $x$, the input by $u$, and the vector field by $f$, which is the standard notation in control theory.

The superposition principle holds for all linear dynamical systems of the form

$$\dot{x} (t) = A (t) x(t) + B (t) u (t), \quad{} x(0) = x_0$$

where $x_0$ is the initial condition, $x : [0,\infty) \to \mathbb{R}^n$ is the state trajectory, $u : [0,\infty) \to \mathbb{R}^m$ is the control input, and $A : [0,\infty) \to \mathbb{R}^{n \times n}$ and $B : [0,\infty) \to \mathbb{R}^{n \times m}$ are matrix-valued functions. This is a system of the form $\dot{x} (t) = f(t,x(t),u(t))$, where the time-varying vector field

$$f : [0,\infty) \times \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$$

is defined by $f (t,x,u) = A (t) x + B(t) u$. Take a look at chapter 5 of Hespanha's book. Are there any other vector fields $f$ for which the superposition principle holds? I don't know.

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