How to set Neumann boundary conditions to solve adjoint variable

variational formulation
,
4

I have explained the mathematical equation, the objective function and adjoint equation i want to solve in this link Calculate sensitivity equation of reaction-diffusion systems with dolfinx and I tried as your tutorial but had this error

import dolfinx 
import numpy as np
import dolfinx.nls.petsc
import ufl
import dolfinx.fem.petsc
import pyvista
from dolfinx.mesh import CellType, create_rectangle
from mpi4py import MPI
from dolfinx import fem, nls, io
from ufl import Measure, inner, grad, div, TestFunction, TrialFunction, adjoint, derivative, SpatialCoordinate, TestFunction, TrialFunction, dot, ds, dx
from dolfinx.fem import (Constant, Function, FunctionSpace,
                         assemble_scalar, dirichletbc, form, locate_dofs_geometrical)
from dolfinx import default_scalar_type
from petsc4py import PETSc
from dolfinx.plot import vtk_mesh


#Parameter
D0 = 0.1;
k0 = -1.0;
eta = 2.0;
gamma = 2.0;


# mesh and function space
mesh = create_rectangle(MPI.COMM_WORLD, [[0, 0], [1, 1]], [100, 100], CellType.triangle)
V = fem.FunctionSpace(mesh, ("CG", 1))
c = fem.Function(V)
phi = TestFunction(V)

# The eps is the design variable
eps = fem.Function(V)
eps.x.array[:] = 0.5

#boundary_markers = FacetFunction('size_t',mesh)

dx = Measure("dx") ##integration over different parts of the domain
x = SpatialCoordinate(mesh)
gu = Constant(mesh, default_scalar_type(0)) ##Neauman = 0

# Residual of the variational form of Poisson problem
R = inner(D0*eps**eta*grad(c), grad(phi))*dx - inner(k0*(1-eps)**gamma*c,phi)*dx - inner(gu,phi) *ds     # " - inner(gu,phi) *ds " is boundary neauman 

##Dirichlet 
def on_boundary(x):
    return np.isclose(x[0], 0)
dofs_L = fem.locate_dofs_geometrical(V , on_boundary)
bc_L = fem.dirichletbc(default_scalar_type(1), dofs_L, V) ##Dirichlet = 1
bcs = [bc_L]

#Forming and solving the linear system
problem_u = fem.petsc.NonlinearProblem(R, c, bcs)
solver = nls.petsc.NewtonSolver(MPI.COMM_WORLD,problem_u)
solver.rtol = 1e-16
solver.solve(c)

J = -ufl.inner((k0*(1-eps)**gamma),c)*dx ##Objective function
J_form = fem.form(J)

dRdc = adjoint(derivative(R, c))
lhs = fem.form(dRdc)
rhs = derivative(J, c)

lmbda = fem.Function(V)
lambda_0 = fem.Function(V)
lambda_0.x.array[:] = 0
bcs_adjoint = [fem.dirichletbc(lambda_0, bc._cpp_object.dof_indices()[0]) for bc in bcs]  
adjoint_problem = dolfinx.fem.petsc.LinearProblem(lhs, rhs, c=lmbda,bcs=bcs_adjoint, petsc_options={"ksp_type": "preonly", "pc_type": "lu"})

Exception ignored in: <function LinearProblem.__del__ at 0x7f334b39d510>
Traceback (most recent call last):
  File "/usr/local/dolfinx-complex/lib/python3.10/dist-packages/dolfinx/fem/petsc.py", line 627, in __del__
    self._solver.destroy()
AttributeError: 'LinearProblem' object has no attribute '_solver'
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In[7], line 12
     10 lambda_0.x.array[:] = 0
     11 bcs_adjoint = [fem.dirichletbc(lambda_0, bc._cpp_object.dof_indices()[0]) for bc in bcs]  
---> 12 adjoint_problem = dolfinx.fem.petsc.LinearProblem(lhs, rhs, c=lmbda,bcs=bcs_adjoint, petsc_options={"ksp_type": "preonly", "pc_type": "lu"})

TypeError: LinearProblem.__init__() got an unexpected keyword argument 'c'