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Merge pull request #793 from JuliaReach/schillic/docs
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Fix formatting in docs page
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schillic authored Feb 28, 2024
2 parents c58a0e6 + fe2a258 commit d11edd8
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10 changes: 5 additions & 5 deletions docs/src/tutorials/linear_methods/dense_time.md
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Expand Up @@ -128,7 +128,7 @@ Algorithms implementing conservative time discretization can be used from the
function.
Set-based conservative discretization of a continuous-time initial value problem
into a discrete-time problem.
This function receives three inputs: the initial value problem (``ivp`) for a
This function receives three inputs: the initial value problem (`ivp`) for a
linear ODE in canonical form, (e.g. the system returned by `normalize`);
the step-size (`δ`), and the algorithm (`alg`) used to compute the approximation model.
Do `subtypes(ReachabilityAnalysis.DiscretizationModule.AbstractApproximationModel)`
Expand All @@ -141,10 +141,10 @@ in the docstring of each method, e.g. [`Forward`](@ref).
Initial-value problems considered in this function are of the form

```math
x' = Ax(t) + u(t),\\qquad x(0) ∈ \\mathcal{X}_0,\\qquad (1)
x' = Ax(t) + u(t),\qquad x(0) ∈ \mathcal{X}_0,\qquad (1)
```
and where ``u(t) ∈ U(k)`` add where ``\\{U(k)\\}_k`` is a sequence of sets of
non-deterministic inputs and ``\\mathcal{X}_0`` is the set of initial
and where ``u(t) ∈ U(k)`` add where ``\{U(k)\}_k`` is a sequence of sets of
non-deterministic inputs and ``\mathcal{X}_0`` is the set of initial
states. Recall that this initial-value problem is called homogeneous whenever `U`
is the empty set. Other problems, e.g. ``x' = Ax(t) + Bu(t)`` can be brought
to the canonical form with the function [`normalize`](@ref).
Expand All @@ -160,7 +160,7 @@ discrete time case.
of the given continuous system are included in the computed flowpipe of the
discretized system. More precisely, given a step size ``δ`` and the system (1)
conservative set-based discretization function computes a set, ``Ω₀``, that
guarantees to contain all the trajectories of (1) starting at any ``x(0) ∈ \\mathcal{X}_0``
guarantees to contain all the trajectories of (1) starting at any ``x(0) ∈ \mathcal{X}_0``
and for any input function that satisfies ``u(t) ∈ U(1)``, for any ``t ∈ [0, δ]``.
If ``U`` is time-varying, this function also discretizes the inputs for ``k ≥ 0``.

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