solution_types.qmd

---
title: "Types of solutions"
bibliography: ref_hybrid.bib
csl: ieee-control-systems.csl
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        html-math-method: katex
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crossref:
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  eq-prefix: Eq.
#engine: julia
---

Now that we know, what a hybrid arc (trajectory) needs to satisfy to be a solution of a hybrid system, we can classify the solutions into several types. And we base this classification on their hybrid time domain $E$:

Trivial
: just one point.

Nontrivial 
: at least two points;

Complete 
: if the domain is unbounded;

Bounded, compact
: if the domain is bounded, compact (well, it is perhaps a bit awkward to call a solution bounded just based on boundednes of its time domain as most people would interpret the boundedness of a solution with regard to the values of the solution);

Discrete 
: if nontrivial and $E\subset \{0\} \times \mathbb N$;

Continuous
: if nontrivial and $E\subset \mathbb R_{\geq 0} \times \{0\}$;

Eventually discrete
: if $T = \sup_E t < \infty$ and $E \cap (\{T\}\times \mathbb N)$ contains at least two points;

Eventually continuous
: if $J = \sup_E j < \infty$ and $E \cap (\mathbb R_{\geq 0} \times \{J\})$ contains at least two points;

Zeno
: if complete and $\sup_E t < \infty$;

Maximal
: It cannot be extended. A solution $x(t,j)$ defined on the hybrid time domain $E$ is maximal, if on an extended hybrid time domain $E^\mathrm{ext}$ such that $E\subset E^\mathrm{ext}$, there is no solution $x^\mathrm{ext}(t,j)$ that coincides with $x$ on $E$. Some literature uses the "linguistic" terminology that a maximal solution is not a prefix to any other solution. Complete solutions are maximal. But not vice versa.

::: {.callout-tip}
It is certainly helpful to sketch the times domains for the individual classes of solutions.
:::

## Examples of types of solutions

::: {#exm-maximal-solution}
## Example of a (non-)maximal solution
$$
\dot x = 1, \; x(0) = 1
$$

$$
(t,j) \in [0,1] \times \{0\}
$$


Now extend the time domain to 
$$
(t,j) \in [0,2] \times \{0\}.
$$

Can we extend the solution?
:::

::: {#exm-maximal-but-not-complete}
## Maximal but not complete continuous solution
Finite escape time

$$\dot x = x^2, \; x(0) = 1,$$

$$x(t) = 1/(1-t)$$ 
:::

::: {#exm-maximal-but-not-complete-2}
## Discontinuous right hand side

$$\dot x = \begin{cases}-1 & x>0\\ 1 & x\leq 0\end{cases}, \quad x(0) = -1$$
(unless the concept of Filippov solution is invoked).
:::

::: {#exm-zeno-solution-bouncing-ball}
## Zeno solution of the bouncing ball

Starting on the ground with some initial upward velocity
$$
h(t) = \underbrace{h(0)}_0 + v(0)t - \frac{1}{2}gt^2, \quad v(0)=1
$$

What time will it hit the ground again?
$$
0 = t - \frac{1}{2}gt^2 = t(1-\frac{1}{2}gt)
$$

$$t_1=\frac{2}{g}$$

Simplify (scale) the computations just to get the qualitative picture: set $g=2$, which gives $t_1 = 1$.

$t_1=1$:

$$v(t_1^+) = \gamma v(t_1) = \gamma v(0) = \gamma$$

The next hit will be at $t_1 + \tau_1$
$$h(t_1 + \tau_1) = 0 = \gamma \tau_1 - \tau_1^2 = \tau_1(\gamma - \tau_1)$$
$$\tau_1 = \gamma$$

$t_2 = t_1+\tau_1 = 1 + \gamma:\quad \ldots$

$t_k = 1 + \gamma + \gamma^2 + \ldots + \gamma^k:\quad \ldots$
$$\boxed{\lim_{k\rightarrow \infty} t_k = \frac{1}{1-\gamma} < \infty}$$


Infinite number of jumps in a finite time!
:::

::: {#exm-water-tank}
## Water tank


![Switching between two water tanks](solution_figures/water_tank.png){width=45%}

$$
\max \{Q_\mathrm{out,1}, Q_\mathrm{out,2}\} \leq Q_\mathrm{in} \leq Q_\mathrm{out,1} + Q_\mathrm{out,2}
$$


![Hybrid automaton for switching between two water tanks](solution_figures/water_tank_automaton.png){width=60%}
:::

::: {#exm-nonblocking-nondeterministic}
## (Non)blocking and (non)determinism in hybrid systemtems
![Example of an automaton exhibitting (non)blocking and (non)determinism](solution_figures/blocking_and_nondeterminism.png){width=60%}

- $x(0) = -3$
- $x(0) = -2$
- $x(0) = -1$
- $x(0) = 0$
:::