# Linear reachability methods

This section begins by introducing the notion of reachable set (reach-set). We show how to visualize and perform set operations with them. Then we introduce flowpipes as the union of reach-sets and illustrate with examples how to perform different visualization and algebraic operations with flowpipes. Then we consider a simple set-propagation problem that consists of a two-dimensional rotation.

After those preliminary sections, we discuss the notion of conservative time discretization for systems of linear differential equations of the form

$$$x'(t) = Ax(t),\qquad x(0) ∈ X_0 ∈ \mathbb{R}^n.$$$

for all times $t ∈ [0, T]$. Linear systems with non-deterministic inputs is discussed in another section of this manual. We also consider an invariant specification: $x(t) ∈ X$ for all times.

The final part of this section introduces support function techniques and discusses the helicopter model application. For ease of exposition, this section only considers. We also show how to introduce state-space invariants, i.e. to impose

## Quickstart guide

The user interface to solve initial-value problems is quite simple.

using ReachabilityAnalysis

# initial-value problem specification
p = @ivp(x' = -x, x(0) ∈ Interval(1, 2))

# flowpipe computation
sol = solve(p, T=5)

Here we have solved the differential equation

$$$x'(t) = -x(t),\qquad x(0) ∈ X_0 = [1, 2] ⊆ \mathbb{R},$$$

for $t ∈ [0, 5]$, whose solution is known to be the decaying exponential $x(t) = x_0e^{-t}$. Let's plot the result, which requires loading the Plots.jl package

variable index 0 denotes time).

using Plots

# post-processing or plotting
plot(sol, vars=(0, 1), xlab="t", ylab="x(t)")

For comparison we can plot the trajectories at the endpoints of the initial interval. The example also shows that the black line segments can be hidden by setting the lw=0 argument (for linewidth).

trange = range(0, 5, length=100)

plot(sol, vars=(0, 1), xlab="t", ylab="x(t)", lw=0)
plot!(trange, 1.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")
plot!(trange, 2.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")

The user-facing interface is designed to be intuitive and interactive, and it is inspired by of other Julia packages such as OrdinaryDiffEq.jl. Moreover, the library internals are written in a modular and composable way, such that advanced users are able to modify and changed easily, or to compose with other algorithms, different steps of the solution process.

In the rest of this notebook we explore different problem specifications, different algorithm choices as well as some processing capabilities. As a preliminary, we introduce the concept of reach-set in the context of the toy model presented above.

## Introducing reach-sets

Basically, a reach-set is a structure that holds a set and a time span associated to it. Here we pick the first reach-set and show that it is of an interval type.

# solutions implement the array interface
R = sol[1]
ReachSet{Float64, Interval{Float64}}(Interval{Float64}([0.94877, 2]), [0, 0.0500001])

Note that sol[1] works since solution structures implement Julia's array interface – meaning that slicing also works, and e.g. sol[end-3:end] returns the last three reach-sets computed.

The solution obtained by set propagation consists of a flowpipe, which is just an array of reach-sets, and behaves like their set union. Flowpipes are at the right level of abstraction concerning solutions obtained with set propagation methods.

We can plot the first reach-set as well.

trange = range(0, 5, length=100)

plot(sol, vars=(0, 1), xlab="t", ylab="x(t)", lw=0)
plot!(R, vars=(0, 1), xlab="t", ylab="x(t)", lw=0, alpha=1., c=:orange, lab="R = sol[1]")
plot!(trange, 1.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")
plot!(trange, 2.0 * exp.(-trange), vars=(0, 1), c=:magenta, lab="")

We can also plot the tenth reach-set in red:

plot!(sol[10], vars=(0, 1), xlab="t", ylab="x(t)", lw=0, alpha=1., c=:red, lab="sol[10]")

The type of reach-sets specifies its numeric type as well as the set representation used; in this case, an interval.

typeof(R)
ReachSet{Float64, Interval{Float64}}

Observing the horizontal axis of the plots reveals that reach-sets have a non-zero along on the time axis. By construction, it is guaranteed that the flowpipe is an enclosure of the true solutions, i.e. no trajectory escapes each reach-set for a given time-span. The time span associated to a reach-set is obtained with the function tspan.

tspan(R)
[0, 0.0500001]

Actually, the algorithm that has been used to solve the problem has a fixed step size of $δ = 0.05$.

tspan(sol[end]) # time-span of the final reach-set
[4.94999, 5.00001]

We can check by extracting the algorithm information from the solution struct:

# methods' step-size
sol.alg.δ
0.05

The functions tstart and tend return the starting and final time of the reach-set.

tstart(R)
-0.0
tend(R)
0.05

The set wrapped by $R$ is obtained with set:

set(R)
Interval{Float64}([0.94877, 2])

We see it is an interval (LazySets.Interval is just a thin wrapper around IntervalArithmetic.Interval; more on this in a note below).

It is interesting to observe that the infimum of sol[1] 0.9487, while we have specified that the initial interval is $X_0 = [1, 2]$. The explanation is that the computed reach-set contains the true solution for all intermediate times between 0 and 0.05 and for all initial states in $X_0$. Since the values of $x(t)$ decrease in the time interval $[0, 0.05]$ the width of sol[1] should be sufficiently big as to enclose those variations. We can make a quick check:

1.0 * exp(-0.05)
0.951229424500714

which shows that R indeed is a correct enclosure of the solution for all points in the initial set. It is easy to check that by decreasing the step-size, the enclosure of the solution at $[0, 0.05]$ converges to the true value (from outside), and similarly for other time intervals.

The time span associated to a reach-sets is an interval satisfying the rules of interval arithmetic. We use IntervalArithmetic.jl to represent time intervals.

ReachabilityAnalysis.TimeInterval # alias
IntervalArithmetic.Interval{Float64}
# retrieve the time span associated to this reach-set
tspan(R)
[0, 0.0500001]
# it is an interval (from IntervalArithmetic.jl)
typeof(tspan(R))
IntervalArithmetic.Interval{Float64}

Time intervals are represented using intervals from IntervalArithmetic.jl. This choice guarantees that all calculations involving time are carried out using rigorous floating-point calculations with interval arithmetic: all quantities are treated as intervals, which are propagated throughout a calculation. The final result is an interval that is guaranteed to contain the correct result, starting from the given initial data.

If desired, it is also work with set using interval elements:

Bint = BallInf(interval.(ones(2)), interval(0.2))
BallInf{IntervalArithmetic.Interval{Float64}, Vector{IntervalArithmetic.Interval{Float64}}}(IntervalArithmetic.Interval{Float64}[[1, 1], [1, 1]], [0.2, 0.200001])

Finally, note that if you create a reach-set by passing a time point, it is automatically converted to an interval:

R = ReachSet(rand(BallInf, dim=5), 1.0)

tspan(R)
[1, 1]