Review ANSI @repl documentation examples with DocumenterVitepress 0.3.5 #904

Context

ct-conventions documents a historical limitation of DocumenterVitepress: before v0.3.5, ANSI-colored output from @repl could be rendered as raw escape sequences. DocumenterVitepress v0.3.5 fixes this behavior (#373), so documentation workarounds may now be obsolete.

Requested review

Please audit the OptimalControl.jl documentation for:

• @repl blocks using try/catch solely to prevent documentation-build failures;
• manual showerror(IOContext(stdout, :color => false), e) calls used only to avoid ANSI rendering issues;
• @example blocks chosen solely because colored @repl output was historically broken;
• opportunities to use direct exception expressions in @repl, relying on REPL-style exception capture;
• cases where @Ansi is more appropriate for an explicit terminal-color demonstration.

Keep @example where regular evaluated-example formatting is preferable, and preserve explicit monochrome output when it is a deliberate presentation choice. Confirm the documentation environment uses DocumenterVitepress >= 0.3.5 and verify the generated site with the documentation build.

Reference implementation

CTBase.jl applied this cleanup in PR control-toolbox/CTBase.jl#547, tracked by issue control-toolbox/CTBase.jl#546. The guidance is in ct-conventions, rules/site.md, notes on the VitePress backend:
https://github.com/control-toolbox/ct-conventions/blob/main/rules/site.md#notes-on-the-vitepress-backend-no-identifier

Please report which blocks were reviewed and update the relevant documentation where appropriate.