[Problems] ModelingToolkitBackend, extension ModelingToolkitModels, and continuous bryson_denham (#308)

Agents: read the guide for agents in #310 first (order, difficulty, how to claim an issue, working method). Read AGENTS.md and clone .ct-conventions (v0.11.0) and .ct-agents (v0.4.0). Specification: this description and #308.

Sub-issue of #308, part 1 of 3. Independent (all prerequisites in #310 are merged).

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Goal

Introduce the ModelingToolkitBackend (:ModelingToolkit) modeling, the ModelingToolkitModels weak-dependency extension, the CTModels.Models component/dimension interface on ModelingToolkit.AbstractSystem, and our own continuous bryson_denham model in ext/ModelingToolkitModels/bryson_denham.jl (with status :draft until Part 2 adds the Corleone equivalence and reference tests), together with problem-retrieval unit tests in test/suite/ModelingToolkit/test_problem.jl.

At the end of this issue, bryson_denham(ModelingToolkitBackend()) is functional, returns a continuous Problem wrapping a ModelingToolkit.System with shared parameters and :init from metadata, and both suite/core and suite/ModelingToolkit pass.

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Technical Findings (from #308 preliminary checks)

1. Continuous path constraint (a) works in ModelingToolkit.System (v11):
   ModelingToolkit.System(eqs, t; costs, constraints, name) accepts a symbolic path inequality depending on the independent variable t = ModelingToolkit.t_nounits:

   cstr = [x₁(tf) ~ x1_tf, x₂(tf) ~ x2_tf, x₁(t) ≲ x1_max]
   

   The time horizon (t0, tf) is carried by the integral cost domain Symbolics.Integral(t in (t0, tf))(0.5 * u(t)^2).
   Therefore bryson_denham(ModelingToolkitBackend(); parameters=nothing, init=:default) takes no grid argument and returns a purely continuous ModelingToolkit.System. (Sampling x₁(t) ≲ x1_max onto constraint_grid is done at solve time in Part 2.)
2. CTModels.Models interface on ModelingToolkit.AbstractSystem:
   Base.show(io, ::MIME"text/plain", pb::Problem) calls CTModels.Models.state_dimension(model), control_dimension(model), and variable_dimension(model).
   In ext/ModelingToolkitModels.jl, define on sys::ModelingToolkit.AbstractSystem:
   • States: unknowns(sys) where !ModelingToolkit.isinput(v)
   • Controls: unknowns(sys) where ModelingToolkit.isinput(v) (declared with [input = true])
   • Variables: parameters(sys) where ModelingToolkit.istunable(p, true) && !ModelingToolkit.isinput(p)
     Strip the (t) suffix from symbolic names so component names are "x₁", "x₂", "u" (matching OptimalControl and JuMP modelings).

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Mandatory Rules & Conventions (ct-conventions v0.11.0, #310)

• Worktree per issue (#310 rule 3a): Never edit or switch branches in the main checkout. Create a dedicated worktree (git worktree add ../OptimalControlProblems-<issue> -b feat/<issue>-mtk-backend main).
• Plan before code (AGENTS.md): Propose a plan following .ct-conventions/plan-template.md and wait for confirmation if working locally with a human, or post the plan on the issue after claiming it (#310 rule 3b).
• Qualified imports only (MOD3, MOD4): Always use using Pkg: Pkg, sym. Never use bare using Pkg or import.
• No top-level exports (MOD8): Only ModelingToolkitBackend is exported from Modelings and re-exported by OptimalControlProblems (matching OptimalControlBackend, OptimalControlScalarizedBackend, JuMPBackend).
• 1-D is a scalar (DIM1): A 1-D state, control, or variable is a Number, not a 1-element Vector.
• No hand-written problem lists (test_no_handwritten_list in test/suite/core/test_consistency.jl): Never write :bryson_denham or "bryson_denham" literals outside docstrings in src/, test/, or ext/*.jl. In tests, always iterate with Main.TestStatus.for_each_problem(..., OptimalControlProblems.ModelingToolkitBackend()) or OptimalControlProblems.problems(OptimalControlProblems.ModelingToolkitBackend()).
• Docstrings (DOC1–DOC8): English only (GEN1), written once implementation is stable.

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Detailed Implementation Plan

###Files to Create or Modify

File	Action	Details
Project.toml	Modify	Add ModelingToolkit = "961ee093-0014-501f-94e3-6117800e7a78" to [weakdeps], ModelingToolkitModels = "ModelingToolkit" to [extensions], and ModelingToolkit = "11" to [compat].
src/Modelings/backends.jl	Modify	Define struct ModelingToolkitBackend <: AbstractModelBackend end. Add (type=ModelingToolkitBackend, name=:ModelingToolkit, ext=:ModelingToolkitModels, pkg="ModelingToolkit") to MODELING_SPECS. Update the docstring table.
src/Modelings/Modelings.jl	Modify	Export ModelingToolkitBackend.
src/OptimalControlProblems.jl	Modify	Import and re-export ModelingToolkitBackend.
ext/ModelingToolkitModels.jl	Create	Module ModelingToolkitModels: include all ext/ModelingToolkitModels/*.jl files; extend CTModels.Models.{state_components,control_components,variable_components,state_dimension,control_dimension,variable_dimension} on ModelingToolkit.AbstractSystem.
ext/ModelingToolkitModels/bryson_denham.jl	Create	Define OptimalControlProblems.bryson_denham(::OptimalControlProblems.ModelingToolkitBackend; parameters=nothing, init=:default): merge parameters with metadata(:bryson_denham)[:parameters], parse init via Problems.parse_init, build the continuous ModelingToolkit.System (@variables x₁(t) = x1_0 [tunable = false] x₂(t) = x2_0 [tunable = false] u(t) [input = true], D(x₁) ~ x₂, D(x₂) ~ u, costs = [Symbolics.Integral(t in (t0, tf))(0.5 * u^2)], constraints = [x₁(tf) ~ x1_tf, x₂(tf) ~ x2_tf, x₁(t) ≲ x1_max]), and return Problems.Problem(:bryson_denham, meta[:title], sys, init_parsed).
ext/MetaData/bryson_denham.jl	Modify	Add ModelingToolkit=:draft to :status.
docs/modelings.jl	Modify	Add :ModelingToolkit => ModelingAdapter(...) to MODELINGS ( required by consistency/doc checks whenever a backend is added to MODELING_SPECS).
test/suite/core/test_modelings.jl	Modify	Add OptimalControlProblems.ModelingToolkitBackend() to BACKENDS.
test/suite/ModelingToolkit/Project.toml	Create	Isolated test environment with ModelingToolkit (compat = "11"), CTBase, CTModels, Test.
test/suite/ModelingToolkit/test_problem.jl	Create	Test Problem retrieval, model(pb) isa ModelingToolkit.AbstractSystem, CTModels dimensions and component names, equations/costs/constraints presence, parameters override, and init variants (:default, :zero, custom NamedTuple, invalid init throwing CTBase.Exceptions.IncorrectArgument) using Main.TestStatus.for_each_problem.

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Verification Commands

julia --project=. -e 'using Pkg; Pkg.test(test_args=["suite/core"])'
julia --project=. -e 'using Pkg; Pkg.test(test_args=["suite/ModelingToolkit"])'

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Done when

• ModelingToolkitBackend is registered in MODELING_SPECS and exported.
• ext/ModelingToolkitModels.jl and ext/ModelingToolkitModels/bryson_denham.jl build a continuous ModelingToolkit.System with continuous path constraint x₁(t) ≲ x1_max, integral cost ∫ 0.5 u(t)², our parameters, and shared :init.
• CTModels.Models dimension and component functions work on ModelingToolkit.AbstractSystem, and Base.show displays the Problem summary cleanly.
• bryson_denham has ModelingToolkit=:draft in ext/MetaData/bryson_denham.jl.
• suite/core (including test_no_handwritten_list and test_modelings) and suite/ModelingToolkit (test_problem.jl) pass with 0 failures/errors.

Refs #308, #310